JimsAstronomy

Monday, December 2, 2019

MATH - How Politicians Use Math?

- 2519 - MATH - How Politicians Use Math? When it comes to math politicians and economists and financial managers do not have a good understanding of reality. They tend to use math to get the numbers they want. Physicists use math and revel at getting numbers they do “not want“. They hope to find what was not expected? Why?
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- ------------------------- 2519 - MATH - How Politicians Use Math?
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- When it comes to math politicians and economists and financial managers do not have a good understanding of reality. They tend to use math to get the numbers they want. Physicists use math and revel at getting numbers they do not want. They hope to find what was not expected? Why?
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- The financial math models currently used on Wall Street grossly oversimplify things. So much so they loose touch with reality. A lot has to do with their use of math as a ‘normal distribution”. This model assumes the events are “ random” in nature.
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- When events are outside a normal distribution they are ignored as outliers, not relevant to what they expect to see. They prefer to view systems in total equilibrium, while the train of thought is going off the cliff. Here are some examples:
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- The 2008 financial crisis, Lehman Brothers, and AIG financial firms go bankrupt (although one was bailed out with our taxes). These firms were using financial models that assessed risk using the Gaussian Curve for a random distribution of the data. This is the same “ Bell Shaped Curve” your teachers used in grade school to grade papers. This always assumed you had a “normal” distribution of students.
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- When these models are presented to the mathematically illiterate it results in a false sense of security. Not only will future results not match reality the unexpected result is “inevitable” . Managers prefer numbers that “ feel good”, so there was no attempts to ask “ why?”
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- The Gaussian distribution of truly random events clusters around an average and tapers off in a predictable way. Above and below the average are “standard deviations” that can be calculated:
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------------------- “1 sigma standard deviations” around the average include 68% of the distribution of events. An event inside this distribution has a 68% chance of happening.
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------------------ “2 sigma standard deviations” around the average include 95% of the distribution.
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---------------- “3 sigma standard deviations” include 99.7% of the distribution. There is only 0.3% risk of the events not occurring inside this distribution around the average. OK, if it is outside 3 sigma it has almost no chance of happening. This is the model that was making elite managers feel good.
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- The Gaussian model says that if an event is 100 standard deviations out it has a probability of 1 chance in 10^350 of occurring. That is a 1 followed by 350 zeros. That event is impossible right?
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- If the math had included all the “outliers” in the data we find that the distribution more accurately follows the “Power Law” distribution , not the Gaussian distribution. The Power Law distribution has much longer tails. Looking at this same data we would say the event had a 1 in 100,000,000 chance of happening, not one in 10^350. If enough of these events occur the chances of seeing this event becomes much more likely. That is the way reality works. Rare events will happen, be prepared.
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- When this “Power Law distribution” was used to predict downturns in the market on the Shanghai Stock Exchange the math predicted the market collapse in 2008 and 2009 within a few days of it actually happening.
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- Power Law math is much messier so it tends to be ignored. Reality is even more complex than this model so it becomes even more difficult. The challenge becomes to figure out the “most important” ingredients to put in the model and deal with those alone.
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- Simple is better but it has to be right. What causes a Power Law to apply to the model? Here is an illustration of how financial leverage creates greater uncertainties.
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- Say you spot an under priced stock. You buy it. Normally a buy would push the price up a bit. Say, you were leveraging on borrowed money to amplify your returns. You borrowed more money to make more purchases. This tends to push the price even higher.
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- Say the bank cuts you off as too high a risk This might force you to sell some of the stock. You might have to sell some other assets prematurely. As prices slide down others might begin selling as well. The amplification cause the tails of the distribution curve to broaden and extend outward with higher and higher risk. The result is a high probability of a crash.
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- This same Power Law analysis can be done with CEO pay, with the number of bird sightings to estimate the population of a certain species, with very large firms affecting the market place, with out-of-control growth, with herding behavior of investors, all of these calculate incorrect results with based on the Gaussian models.
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- Power Law models are an improvement but even that complexity is not enough. Thoughtful analysis of what is behind the numbers is needed. Are the important ingredients part of the model? Are outliers part of reality that need to be included?
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- Math is a tool. You need to know how to use it. All models are an approximation of reality. Expect the unexpected to happen.
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- December 2, 2019 2518 1324
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----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Monday, December 2, 2019 --------------------
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MATH - Interesting tricks and shortcuts?

- 2518 - MATH - Interesting tricks and shortcuts? Was math discovered in nature, or, was it invented to describe nature. Regardless, it is amazing that it works so perfectly. Here are some mathematical patterns that I will show how they work.
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-------------------- 2518 - MATH - Interesting tricks and shortcuts?
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- Science and Math are special to me. Science is the study of reality. It helps me try to understand what the world is really like. Math is the study of patterns. What is amazing to me is how well math is used to describe nature. Was math discovered in nature, or, was it invented to describe nature. Regardless, it is amazing that it works so perfectly.
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- If you multiply two of the same numbers together you will always get the maximum product for their sum.
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--------------- 5 * 5 = 25 their sum is 10. If you take 4 * 6 you get 24, one less than the maximum. If you take 3 * 7 you get 21 which is 4 less than the maximum, or 2^2 less than the maximum. Try this by squaring any number:
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-------------- 10 * 10 = 100
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-------------- 9 * 11 = 99 --------------- one less than max ----- sum is 20
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-------------- 8 * 12 = 96 --------------- two^2 less than max ----- sum is 20
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-------------- 7 * 13 = 91 --------------- three^2 less than max ----- sum is 20
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- So, the square of any number becomes easier to calculate. Here’s how:
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-------------- 13 * 13 = 169
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-------------- 12 * 14 = 168 --------------- one less than max ----- sum is 26
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-------------- 11 * 15 = 165 --------------- two^2 less than max ----- sum is 26
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-------------- 10* 16 = 160 ---------------
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- So, to square 13, multiple 10*16 = 160 , a much easier calculation to do in your head, and, add 3^2, or 9 to get 169.
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-------------- 13 * 13 = 169
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--------------------------- To square 15, 10*20 = 200 + 5^2 = 225
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--------------------------- To square 21, 22 * 20 = 440 + 1^2 = 441
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- This works for the square of any number.
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- Carl Freidrich Gauss was a math genius at 10 years old. When he was in the 4th grade the teacher tried to keep the class busy by asking them to add up all the numbers between 1 and 100. Gauss immediately blurted out the correct answer 5,050.
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- Gauss did it in his head by taking the long string of numbers 1,2,3, …… 99,100 and folding them back on themselves to create pairs of numbers 1+100, 2+99, 3+98, ….. 50+51. It was easy for him to see that the pair always added up to 101 and this happened just 50 times. So, 101 * 50 can be done in your head, it’s 5,050.
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- The sum of any sequence of patterned numbers is n*(n+1) / 2.
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- Where “n” is the last number in the sequence. Summing the numbers up to 100 is:
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-------------------- 100 (101) / 2 = 10,100 / 2 = 5,050
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- If you did the same problem adding up all the numbers from 1 to 500. Now “n” is 500:
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-------------------- 500 (501) / 2 = 250,500 / 2 = 125,250
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- The formula changes a little if you add up the pattern of all the even numbers to the number “n“, 2+4+6+8+10+……… 100. Where n = 100.
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-------------------- n(n+1)
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----------- 100 ( 101 ) = 10,100 , which is the sum of all the even numbers up to 100.
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- The formula changes to n^2, if you add up the pattern of all odd numbers to the number “n”. Adding up all the odd numbers from 1 to 100 = 100^2 = 10,000.
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- An interesting pattern exists between the squares of numbers and the cubes of numbers. It turns out that the sum of all the cubes of numbers up to “n” is the square of the sum of the numbers up to “n”.
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------------- For example: The sum of all the cubes up to 5 is:
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------------- 1^3 + 2^3 + 3^3 + 4^3 + 5^3 =
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------------- 1 + 8 + 27 + 65 + 125 = 225
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------------- And, (the sum of all numbers up to 5) squared is:
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------------- ( 1 + 2 + 3 + 4 + 5 )^2 = 15^2 = 225
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------------- This holds for all numbers up to any number.
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- Another interesting pattern in squaring numbers is that if number ends in 5 the square always ends in 25.
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----------------- (35)^2 = 1225
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----------------- You can do this calculation in your head because what is also true is that the answer is always preceded by the first number multiplied by one higher than that number. The first number of the number 35 which is being squared is 3. The one higher is 4. 3 * 4 = 12. and, the answer always ends in 25 so the answer is 1225. This works for the square of all numbers ending in 5:
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------------------ (35)^2 = 3*4 + 25 = 1,225
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----------------- (55)^2 = 5*6 + 25 = 3,025
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----------------- (75)^2 = 7*8 + 25 = 5,620
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----------------- (105)^2 = 10*11+ 25 = 11,025
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- If your 4th grade teacher asks you how to add up all the numbers in the multiplication table up to 100 here is how you do it: You start with a 10 by 10 matrix of row and columns that contain the multiplication table. The first row is 1 through 10 and you already know that sum to be n(n+1)/2, remember.
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------------ For the first row the sum is 10*11/2 = 55
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------------ For the second row the sum is 55 * 2 = 110
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------------ For the third row the sum is 55*3 = 165
……………………
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------------ For the tenth row the sum is 55 * 10 = 550
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- So, you can see that the sum is going to be 55*1 + 55*2 + 55*3 + ……..
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- And, you can factor out the 55 and write it as 55*(1 +2 +3 +4 ……..)
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- And, you already know what the sum of the second factor is, it’s 55.
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- Therefore, the sum of all the numbers in the multiplication table up to 100 is 55*55.
55^2 is another thing you already know. How to square any number ending in 5.
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------------------------------- 55^2 = 5*6 ….. 25 = 3,025.
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- If you sum up all the numbers in the multiplication table up to 100 you get 3,025.
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- You learned that squaring any number ending in 5 is always 25 preceded by the first digit times one greater and putting it in front of the 25:
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------------------------ 35^2 = 3 * 4 in front of 25 = 12 …. 25 = 1,225
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------------------------ 55^2 = 5 * 6 ………. 25 = 3,025
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- You can do these multiplications in your head.
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- Actually, the rule being applied here works for squaring any number and, in fact, for multiplying any 2 numbers. To use fancy words for a moment, in algebra it is multiplying polynomials together to get polynomials of the second degree. Algebra uses the quadratic equation to solve polynomial equations of the second degree. None of this algebra do you need to learn in order to perform these tricks in multiplication.
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----------------- 55^2 = (50 + 5)(50+5) = 50( 50+5+5) + 5^2 = 50 * 60 + 25 = 3,025.
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- The algebraic way to say this is (a+b)(a+b) = a^2 +ab +ab + b^2 = a(a+b+b) + b^2.
Now that looks complicated but you will be surprised how easy this algebra is to use after some practice.
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--------------- 95^2 = (90 + 5) (90 + 5) = 90 (100) +25 = 9,025
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- This process does not just work with squaring numbers. It works multiplying any two numbers. The idea is to change the multiplication so it contains easy numbers to multiply in your head.
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------------------ 23 * 28 = (20 + 3)( 20 + 8) = 20 * (20+3+8) + 8*3 = 20 * 31 + 24 = 620 + 24 = 644.
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- The general algebraic formula is (a+b)(c+d) = ac +ad + bc + bd. When you make the first number term the same it becomes (a+b)(a+c) = a^2 + ac + ab + bc = a ( a+b+c) + bc.
Remember these letters in algebra are just substitutes to stand for any number. They call them variables. But, once you choose what a letter stands for than the math is fixed for that result. a,b,c in this case are 100, 7,and 11:
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------------------ 107 * 111 = (100 + 7)( 100 + 11) = 100 ( 100 + 7 + 11) + 7*11 =
100 ( 118) + 77 = 11800 + 77 = 11,877
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-------------------94 * 91 = (100 -6 )(100 - 9) = 100 * (100 - 15) + (-6)(-9) = 100 * 85 + 54 = 8500 + 54 = 8,554.
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- It takes some practice, but when you get the hang of it many complicated multiplications can be made simpler using this algebra trick.
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----------------- 62 * 68 = (60 + 2)( 70 - 2) = 4200 + 140 - 120 - 4 = 4200 +20 - 4 = 4216, or even easier:
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----------------- 62 * 68 = (60 + 2)( 60 + 8) = 60 ( 60+2+8 ) + 16 = 60 ( 70) + 16 = 4200 + 16 = 4,216
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- You can quickly multiply a 2-digit number together by adding them and sticking the sum between the two numbers.
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--------------- 23 * 11 = 253
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- If the sum is greater than 9 remember to carry the one:
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-------------- 96 * 11 = 1,056
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- You can do it with numbers having more than 2 - digits by adding the numbers 2 at a time and sticking their sums between the first and last numbers:
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---------------- 423 * 11 = 4, 653
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- There are many tricks you can play with algebra. Here is another one for you teachers:
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- You ask your students to pick any 4-digit number .
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--------------------------- 1,618
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- Ok, now scramble the number to make a different number.
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--------------------------- 8,611
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- Subtract the smaller number from the larger number.
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--------------------------- 6,993
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- Now add up the digits in that number.
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--------------------------- 27
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- If you have a one digit number you are done. If you have a 2 digit number add those up.
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--------------------------- 9
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- The answer you get is 9. Right? The answer you get will always be 9. Trust me.
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----------------------------------- 4362
----------------------------------- 6234
----------------------------------- 1872
----------------------------------- 18
----------------------------------- 9
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----------------------------------- 9721
----------------------------------- 2791
----------------------------------- 7002
----------------------------------- 9
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Practice these tricks and they will become easy. And, you will have confidence in the results. Let me know if you have any questions. Review that comes next teaches you how to predict the day of the week for any date of the year. Your birthday for instance. This is even more interesting. It is called modular math.
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- December 2, 2019 2518 796 798
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----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Monday, December 2, 2019 --------------------
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Sunday, December 1, 2019

JOKE - interesting facts

- Interesting facts so we have something to talk about over coffee:
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- In the 1400's a law was set forth in England that a man was allowed to beat his wife with a stick no thicker than his thumb. Hence we have "the rule of thumb."
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Many years ago in Scotland , a new game was invented. It was ruled "Gentlemen Only...Ladies Forbidden"...and thus the word GOLF entered into the English language.
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The first couple to be shown in bed together on prime time TV were Fred and Wilma Flintstone.
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Every day more money is printed for Monopoly than the U.S. Treasury.
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Men can read smaller print than women can; women can hear better.
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Coca-Cola was originally green.
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It is impossible to lick your elbow.
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The State with the highest percentage of people who walk to work: Alaska
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The percentage of Africa that is wilderness: 28% (now get this...)
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The percentage of North America that is wilderness: 38%
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The cost of raising a medium-size dog to the age of eleven: $16,400
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The average number of people airborne over the U.S. in any given hour: 61,000
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Intelligent people have more zinc and copper in their hair. I can hardly comb mine.
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The first novel ever written on a typewriter: Tom Sawyer
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The San Francisco cable cars are the only mobile National Monuments.
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Each king in a deck of playing cards represents a great king from history:
Spades - King David; Hearts - Charlemagne; Clubs - Alexander, the Great; Diamonds - Julius Caesar
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111,111,111 x 111,111,111 = 12,345,678,987,654,321
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If a statue in the park of a person on a horse has both front legs in the air, the person died in battle. If the horse has one front leg in the air the person died as a result of wounds received in battle. If the horse has all four legs on the ground, the person died of natural causes.
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Only two people signed the Declaration of Independence on July 4th, John Hancock and Charles Thomson. Most of the rest signed on August 2nd, but the last signature wasn't added until 5 years later.
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Q. Half of all Americans live within 50 miles of what?
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A. Their birthplace
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Q. Most boat owners name their boats. What is the most popular boat name requested?
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A. Obsession
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Q. If you were to spell out numbers, how far would you have to go until you would find the letter "A"?
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A. One thousand
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Q. What do bulletproof vests, fire escapes, windshield wipers, and laser printers all have in common?
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A. All were invented by women.
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Q. What is the only food that doesn't spoil?
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A. Honey
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Q. Which day are there more collect calls than any other day of the year?
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A. Father's Day
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In Shakespeare's time, mattresses were secured on bed frames by ropes. When you pulled on the ropes the mattress tightened, making the bed firmer to sleep on. Hence the phrase "goodnight, sleep tight."
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It was the accepted practice in Babylon 4,000 years ago that for a month after the wedding, the bride's father would supply his son-in-law with all the mead he could drink. Mead is a honey beer, and because their calendar was lunar based, this period was called the honey month, which we know today as the honeymoon.
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In English pubs, ale is ordered by pints and quarts. So in old England when customers got unruly, the bartender would yell at them, "Mind your pints and quarts, and settle down." It's where we get the phrase "mind your P's and Q's"
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Many years ago in England , pub frequenters had a whistle baked into the rim, or handle, of their ceramic cups. When they needed a refill, they used the whistle to get some service. "Wet your whistle" is the phrase inspired by this practice.
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~~~~~~~~~~~AND FINALLY~~~~~~~~~~~~
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At least 75% of people who read this will try to lick their elbow!
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Don't delete this just because it looks weird. Believe it or not, you can read it.
I cdnuolt blveiee taht I cluod aulaclty uesdnatnrd waht I was rdanieg. The phaonmneal pweor of the hmuan mnid. Aoccdrnig to rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is taht the frist and lsat ltteer be in the rghit pclae. The rset can be a taotl mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. Amzanig huh?
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- December 1, 2019.
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----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Sunday, December 1, 2019 -------------------------
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CALCULUS - Spacetime and Energy?

- 2515 - CALCULUS - Spacetime and Energy? The theory of relativity teaches us that space and time are relative. Both are always relative to some reference plane. So, keep that in mind even though most of this review will be strictly Newtonian. In Einstein’s world space ,”s”, collapses and time,”t”, slows down as velocity approaches the speed of light, “c”.
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-------------------- 2515 - CALCULUS - Spacetime and Energy?
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- Velocity, c = s / t. Velocity of light is meters / second, 300,000,000 meters/second. And, “c” is constant, regardless of the relative motion of objects. Therefore, if an object in space collapses like a pancake as it approaches the speed of light, “s” becomes smaller, and in order to keep “c” constant, time must get smaller too, fewer clicks of the clock.
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- Therefore, clocks run slower as you go faster and faster, yes, that includes biological clocks. NASCAR drivers are younger than their twin brothers the accountants . Using Einstein’s equations you can actually calculate how much younger. But, I said this review would be Newtonian, actually it is whole course in Physics:
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------- SPACE is measured as distance or length and the units are meters, symbol is “s“.
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-- TIME is measured as duration or elapsed time and the units are seconds, symbol is “t“.
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---------------------------------- FOR OBJECTS WITH MASS:
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- VELOCITY is the rate of change of distance with time. v = d(s) / dt
Velocity is a change of space over a change of time. It is the derivative of space with respect to time.
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- If you plot two variables distance versus time, the slope of the line on the graph is velocity, it is the rate of change of distance with respect to time. Calculus calls it a “derivative“, but you could just as easily call it a “rate of change” or a “slope” of one variable versus another. The units are meters / second.
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- ACCELERATION is the rate of change of velocity with time a = d(v) / dt
Again, the little “d” refers to calculus and derivatives.
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- Acceleration is the second derivative of space with respect to time and the first derivative of velocity with respect to time. a = d^2(s) / dt^2
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- The units are meters / second / second, or, m / sec^2
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- The integral of acceleration is velocity. v = integral a (t)
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- INTEGRATION in CALCULUS is simply “ summation”. the summation of little areas. If you plot acceleration versus time, the y-axis is “a“, the x-axis is “t“. Then an area on the graph is “a * t” , a rectangle area is length times width. And, a = s / t^2, that is acceleration is meters/sec^2. And s / t^2 * t = s / t = velocity.
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- So, if you know acceleration and you integrate you get velocity. And, if you know velocity and you differentiate you get acceleration. That is almost all there is to CALCULUS, simple huh?
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-------------------------------- ENERGY = mass * velocity^2
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------------------------------- E = m*c^2 = m * (9*10^16)
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------------------------------- The Energy of motion, KE = kinetic energy = ½ m* v^2
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------------------------------- Units of mass are kilograms
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------------------------------- Units of Energy are kg * m2 / sec^2
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- Energy is mass times the (first derivative of space versus time) squared. Energy can neither be created nor destroyed. Energy is Conserved. We have the same amount of Energy today as we did with the Big Bang.
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- With Relativity E = 1/(1-v^2 / c^2)^½ * m * c^2. The new factor that is added, 1/(1-v^2 / c^2)^½ is called the Lorentz Factor. With this factor the amount of energy that is required to accelerate an object to the speed of light becomes infinite. Therefore, it can’t be done.
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------------------------- Energy is also the capacity to do Work.
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------------------------- ENERGY = E = m*v^2 = p*v = m * (ds / dt)^2 = A/t
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----------------------- WORK is a change in Energy, W = E1 - E2.
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------------------------- Work is the product of a Force and distance, W = F * s
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----------------------- Units of Work are the same as units of Energy, kg*m^2 / sec^2
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-------------------- MOMENTUM is a product of mass and velocity, p = m * v
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-------------------- Units for momentum are kilogram * meters / sec
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-------------------- Energy is momentum * velocity, E = p * v
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-------------------- Units of Energy are kg* m^2 / sec^2
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- Energy is the momentum times the rate of change of distance with respect to time. Remember, every time you see “ rate of change” “with respect to” you can replace that with “derivative“. E = p*ds/dt
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- Momentum is the ratio of Energy / velocity. p = E /v
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- With Relativity momentum = p = 1/(1-v^2 / c^2)^½ * m*v. An object can not be accelerated to the speed of light because its inertia goes to infinity as “v“ approaches “c“.
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--------------------- FORCE is a product of mass and acceleration. F = m * a
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-------------------- Force = mass * d(v) / dt
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-------------------- Force = mass * the rate of change of velocity with respect to time.
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-Force = mass * the 2nd derivative of distance with respect to time, F = m * d^2(s) / dt^2
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-------------------- Force = kg * m / sec^2
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--------------------- Energy is Force operating over a distance. E = F * s
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-------------------- Force = Energy / distance. F = E / s.
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-------------- Force = the rate of change of momentum with respect to time. F = d(p) / dt
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-------------------------------- ACTION is a product of Energy and time. A = E * t
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-------------------- Units of Action are = kg * m^2 / sec
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-------------------- Energy = Action per unit period of time, kg*m^2/sec / sec
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-------------------- Energy is the rate of change of Action over time, E = A / t
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-------------------- Action is the product of Momentum and distance, (kg*m/sec )* meters
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-------------------- A = p*s
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-------------------- Action = Force * distance * time . Units are kg * m^2 / sec
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-------------------- Action = mass * velocity^2 * time . Units are kg * m^2 / sec
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-------------------- Action = F*s*t = m*a*s*t = m*v^2*t = E * t = m*v*s = p * s
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- Acceleration is the derivative of velocity with respect to time. It is the slope of the line of velocity plotted versus time. INTEGRATION in CALCULUS is somewhat the opposite of derivatives.
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- The integral of acceleration with respect to time is velocity,
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------------------------------ v = integral a * dt , a = d(v) / dt
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- Distance is the integral of velocity with respect to time, s = integral v * dt, v = d(s) / dt
Action is the integral of momentum with respect to time, Action = integral p * dt,
p = d(A) / dt
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- WORK is the integral of force with distance, Work = integral F * ds. Work is equal to Force * distance, if the force is constant. But, if the force is varying over the distance you have to use calculus and take the integral of F * ds.
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------------------------------- FOR MASSLESS OBJECTS
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- PHOTONS have no mass, so E = mc^2 will not work. E = 0. But, the energy of a photon is not zero. We know photons have energy. We know photons have momentum.
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- The classic demonstration I remember is the black and white vanes inside a clear light bulb. When a light is shined on the blades the black sides absorb the light and the white sides reflect the light and the net rotational force causes the blades to spin inside the light bulb.
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- We can still use the formula E=m*c^2 if we make some changes. m = E / c^2
m = E / c*c
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-------------------- m*c = E/c
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-------------------- m*c is mass times velocity and that is momentum, so, p = E/c.
- So, the energy of a photon is equal to its momentum times the speed of light.
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- We also know that the energy of a photon increases with its frequency. From radio waves to X-rays and Gamma rays the energy of photons increase with frequency.
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- The breakthrough came in 1905 when Einstein and Planck determined that light energy was not continuous. It came in packets, bundles of energy, called quanta, and later called photons. Each photon had a fixed amount of energy.
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- ENERGY operating over time is called “ Action”. A = E * t
And, each photon has a fixed amount of action. It is called Planck’s Constant of Action, and it has the symbol, “h”.
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------------------------------- h = 6.6 * 10^-34 kilogram*m^2 / sec
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----------------------------------------- ACTION = A = E*t
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-------------------- h = the action of one photon
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-------------------- Energy of each photon depends on its frequency = E = h * f
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-------------------- E = 6.6*10^-34 * f
-
-------------------- E = 4.136*19^-15 electron volts * f
-
-------------------- “c” = wavelength * frequency, meters / second.
-
-------------------- c= w * f
-
-------------------- E = h * c / w
-
-------------------- E = mc^2 = m*c*c = p*c, kg*m^2/sec^2
-
--------------------- ENERGY of a photon = momentum * c, E = p*c
-
-------------------- E = m*c^2
-
-------------------- E = m*c^2 / (1-v^2 / c^2)^½ as velocity approaches speed of light.
-
-------------------- Energy of a photon = h * f
-
-------------------- Energy of a photon = 6.63 * 10^-34 * frequency, kg * m^2 / sec^2
-
-------------- ENERGY = E = h*c / w = p*c / w = h*f = 6.6*10^-34 * f = 3*10^8 * p
-
- VELOCITY of electromagnetic radiation is a constant. The symbol is “c”
The velocity of a photon = wavelength * frequency. c = w * f = 3*10^8 m/s
Units for a wavelength = meters / cycle
-
-------------------- Units for frequency = cycles / second
-
-------------------- Units for velocity is meters / second
-
-------------------- “c” = 299,792,458 meters / second, or, approximately = 3*10^8 m/sec
-
- MOMENTUM of a photon = Planck’s Constant of Action / wavelength, p = h / w
Momentum of a photon = 6.63 * 10^-34 kg * m / sec.
-
- The momentum of a light wave increases as the wavelength decreases, or momentum increases as the frequency of the light increases.
-
- An example using these equations. Comparing the energy of yellow light to ultraviolet light, sunlight to sun tans.
-
----------------------------- E = h*c/w w for yellow photons is 600 nanometers.
-
-------------Yellow E = 6.6*10^-34 * 3*10^8 / 6*10^-10 = 3.3 * 10^17 kg*m^2/sec^2
-
------------------------Ultraviolet light photons have a wavelength of 400 nanometers.
-
----- Ultraviolet E = 6.6*10^-34 * 3*10^8 / 4*10^-10 = 4.95 * 10^17 kg*m^2/sec^2
-
---------------------Therefore Ultraviolet light has 50% more energy than yellow light.

----------------------- p = h/w = h*f / c = E /c = E / 3*10^8 kg * m/sec
-
- WAVELENGTH of a photon = w = c / f = h*c / h*f = h*c / E = h / p, meters
-
- FREQUENCY = E / h = w / 2*pi = 3*10^8/w =
-
- NUMBER of PHOTONS = Total Energy / h*f per photon, (See Review 2113)
-
---------------------------------- FOR QUANTUM SIZED OBJECTS:
-
- The smallest unit of acton = 6.6*10^-34 kg*m^2/sec
-
- The smallest unit of mass = 1.1*10^-8 kg, m^2 = h*c/8*G*pi
-
- The smallest unit of length = 1.6163* 10^-35 meters, s^2 = G*h/2*pi*c^3
-
- The smallest unit of time = 1.4*10^-43 sec., t^2 = G*h/2*pi*c^5
-
- The reason that there is a smallest unit of length is that the mass would turn into a black hole if the diameter, or length, got any smaller. And the reason that there is the smallest unit of time is that is the time it takes light to travel across that smallest unit of length. (See Review 724)
-
- See Review 607 - “Antimatter is all around us”
-
------------------------ Unit of Energy = joules = kg*m^2/sec^2
------------------------ Unit of Force = newtons = kg*m/sec^2
-
- December 1, 2019 2515 734
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
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--------------------- Sunday, December 1, 2019 --------------------
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CALCULUS - Calculus of a Circle?

- 2516 - - CALCULUS - Calculus of a Circle? Calculus is the mathematical concept of doing Differentiation and Integration. For example, Integration is done by dividing an object up into many, many pieces and adding these up to get the whole object. This concept of summation is called Integration. Differentiation is a rate of change, or the slope of rise over run with one variable compared to another.
-
-

-
--------------------- 2516 - CALCULUS - Calculus of a Circle?
-
- How did the ancients define the area of a circle to be pi*r^2? Well, first of all they had no TV’s, so they did not have to watch American Idol, or Dr. Phil, or Entertainment Tonight. They had no electricity so they sat by candlelight and had time to think.
-
- They figured out that all circles , regardless of their size, had the same ratio of circumference to diameter. Let’s call it c / d.
-
- And the circumference of any circle must be circumference = d*c / d = c.
-
--------------------- Or, c = 2*r*c / d.
-
- Let’s take a circle and divide it into wedges just like you would cut up a pizza pie. Now take each wedge and stack them side by side with a point up, then a point down, then up, then down, etc. until you have formed a parallelogram.
-
- If you cut the pie in small enough wedges and stack them this way it will look more and more like a rectangle. And the area of a rectangle is length times height. Cut the pie into a million pieces and you have a rectangle for sure. Area = length * height.
-
------------------------ But, the height is radius, r.
-
------------------------ And, the length is half the circumference, ½*c.
-
------------------------ The circumference is 2*r*c/d.
-
------------------------ So, the Area = r*½* 2*r*c/d
-
------------------------ Or, Area = c/d * r^2
-
- OK, now all we have to do is define c / d as pi, after our pizza pie.
And, Area = pi*r^2
-
- Summing up all these pieces became known as integration and Calculus. And, you can do amazing things with the mathematics of Calculus. Pi became calculated as 3.14159.….. the calculation goes to infinity.
-
- If you take the derivative of the area of a circle you get the circumference,
d(pi*r^2) /dr = 2*pi*r ( in the math tables it is d(x^n) = n*x^(n-1)* dx).
-
- So, like the velocity is the rate of change of distance with respect to time. d(v) = dx/dt The area is the rate of change of the circumference with respect to the radius. d(A) = dc/dr
The integral of the circumference with respect to the radius is the area.
-
- The intergal of 2*pi*r = 2*pi*r^2 / 2 = pi*r^2. ( in the math table the integral of x^n*dx = x^(n+1) /( n+1)
-
- This makes sense, it you start with a circle with 1 inch radius you get a circumference of 6.28 inches. And it is a hollow circle. But, if you start with a point and you integrate, or do a summation, of each circumference out to 1 inch radius you get a solid circle with an area of 3.14 square inches.
-
- This same process of derivatives and integrals works for the volume of a sphere, 4/3*pi*r^3, to the surface area of a sphere, 4*pi^r^2.
-
- TO SUMMARIZE:

------------------ The perimeter of a circle is the rate of change or area with radius. Rate of change is the slope, or the derivative of area with radius. The derivative of pi*r^2 = 2*pi*r. The general formula for the derivative of x^n * dx = x^n-1
-
----------------- The area of a circle is the summation of the perimeters with radius. Summation is integration of perimeter with radius. The integration of 2*pi*r = p*r^2.
The general formula for the integration of x*dx = x^n+1 / n+1
-
------------------ The surface area of a sphere is the rate of change or volume with radius. Rate of change is the slope, or the derivative of volume with radius. The derivative of 4/3*pi*r^3 = 4*pi*r^2. The general formula for the derivative of x^n * dx = x^n-1
-
----------------- The volume of a sphere is the summation of the surface areas with radius. Summation is integration of surface areas with radius. The integration of 4*pi*r^2 = 4/3*pi*r^3. The general formula for the integration of x*dx = x^n+1 / n+1
-
- Pi is an irrational number, which means it can not be written as a ratio of two integers. 22/7 is only accurate to 2 decimal points. 355/113 is accurate only to 6 decimal places.
-
- In 1657 pi originally stood for “peripheral / diameter” ( I was just kidding about the pizza pie). In 1600 Ludolph van Ceulen calculated pi out to 35 decimals. He was so proud of what he had done he had it engraved on his tombstone. The symbol for pi was first used in 1706.
-
- In 1789 pi was extended out to 140 places ( of which it was later found that only 137 places were correct). In 1873 it was extended to 707 places and that stood until 1949 when the computer was invented. Today it is out to 1,240,000,000 places.
-
- 3.1415926535897932384626433832795028841971.……… to infinity
-
- You can remember some of these decimal places by remembering “ How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics” = 3.14159265358979. Count the letters and you have pi to 14 decimal points.
-
- Pi shows up in a lot of places. See Review 513 “Buffon’s Needles”, which tell how repeatedly dropping breadsticks on a tile floor can be used to statistically calculate the value of pi.
-
- See Review 400 “Life is Uncertain“, Heisenberg’s uncertainty principle for delta position * delta momentum = Planck’s constant / 4*pi.
-
--------------------- Or, Einstein’s field equation: R-1/2gR + delta g = 8*pi*G^4*T.
-
---------------------- The series 1/1^2 + ½^2 + 1/3^2 + … = pi^2/6
-
---------------------- The series 1/1-1/3+1/5-1/7+1/9-…… = pi/4
-
---------------------- e^i*pi +1 = 0
-
- The great pyramids in Egypt were constructed with the ratio of the perimeter of the base to twice the height = pi.
-
- How all these things have to do with circles I can not imagine. But the ratio of the circumference to the diameter sure shows up in a lot of places. It all must be connected somehow???
-
- Other Reviews available upon request:
-
- 513 “Buffon’s Needles”, which tell how repeatedly dropping breadsticks on a tile floor can be used to statistically calculate the value of pi.
-
- 400 “Life is Uncertain“, Heisenberg’s uncertainty principle for delta position * delta momentum = Planck’s constant / 4*pi
-
- 2514 - Everyday Calculus
-
- 734 - Spacetime, Energy, and Calculus
-
- December 1, 2019. 2516 745
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
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--------------------- Sunday, December 1, 2019 -------------------------
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CALIFORNIA - California’s Mega-floods?

- 2515 - CALIFORNIA - California’s Mega-floods? Every 200 years California experiences a mega-flood from the rivers in the sky. Our last enormous flood occurred in the year 1861.
-
-

-
-------------------- 2515 - CALIFORNIA - California’s Mega-floods?
-
- It is raining, and it is expected to continue to rain for many days in a row. I am starting to worry. Every 200 years California experiences a mega-flood from the rivers in the sky. Our last enormous flood occurred in the year 1861.
-
- The Central Valley of California became a lake 300 miles long and 20 miles wide. 800,000 cattle drowned. Downtown Sacramento was under 10 feet of brow water. The Legislature had to move to San Francisco for 6 months before it could return. The state went bankrupt, much like it is today. Although the bank rupture was due to politics, not rain.
-
- Today the Central Valley is home to 6,000,000 people that would have to be evacuated in such a flood. 1,400,000 live in Sacramento. Since 1861 the valley floor has sunk 30 feet due to extensive ground water pumping. Flooding would be much worse this time. That land produces $20 billion in crops annually. Estimates are that a smaller 23 day storm would cause $400 billion in property damage and agriculture losses.
-
- Sediment cores and radioactive dating have identified major floods in Northern California going back to 212 AD:
-
-------------------------------212 AD
-----------------------------------------------------------220 years
-------------------------------440 AD
----------------------------------------------------------163 years
-------------------------------603 AD
---------------------------------------------------------- 426years
-------------------------------1029 AD
---------------------------------------------------------- 71years
-------------------------------1100 AD
----------------------------------------------------------300 years
-------------------------------1400 AD
----------------------------------------------------------250 years
-------------------------------1650 AD
----------------------------------------------------------211 years
-------------------------------1861 AD
----------------------------------------------------------152 years and counting
-------------------------------2013 AD
--------------------------------------------------------AVERAGE: 234 years
-
- The 1100 to 1650 dates came from marsh deposits around San Francisco
-
- These enormous floods are caused by “ atmospheric rivers”. One mile high. 250 miles across. Extending for thousands of miles. They can carry as much water as 15 Mississippi Rivers. When they are smaller rivers they get the name” pineapple express”. About 9 of these hit California during some of our winter.
-
- The rivers are created in the atmosphere by warm, moist air of the tropics rising into the cooler, dry regions and like a conveyor delivery water to Northern California.
-
- The good news is that these atmospheric rivers supply 30 to 50 percent of California’s water in the span of 10 days. They also cause 80% of the flooding. So, they are vital to California’s water supply, but, we pay a price for it.
-
- Computer models show that a smaller 23 day rain storm will cause more devastation to the state than a 7.8 earthquake. It would cost California’s economy $700,000,000,000.
-
- We have 37,691,912 people
-
- We have 15,800,000 people who file taxes, some pay nothing even though they file.
-
- Each taxpayer would have to pay $44,300 to recover the damages from the storm, for what could be recovered.
-
- He who ignores history is destined to repeat it. An announcement will be made shortly, stay tuned. Our Coffee Club will be working on the problem.
-
- To learn more:
-
---------------- National Oceanic and Atmospheric Administration, NOAA
--------------- www.esrl.noaa.gov/psd/atmrivers
-
--------------- US Geological Survey USGS
-------------- ARkStorm: http://urbanearth.gps.caltech.edu/winter-storm-2
-
- December 1, 2019 2515 1544
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Sunday, December 1, 2019 --------------------
-----------------------------------------------------------------------------------------

CALCULUS - Everyday Calculus

- 2514 - CALCULUS - Everyday Calculus. Calculus does only two things. It is not complicated. But, academics make it so, Sorry! All my math professors were in the weeds. They had no clue how to teach a subject. They tried to make it complicated in order to intimidate you and make themselves feel important. If you make it simple it is simple to understand.
-
-

-
-------------------- 2514 - CALCULUS - Everyday Calculus
-
- If you make it simple it is simple to understand. “You can understand if you care”. The two things Calculus does is Differentiation and Integration. These are big words, but very simple ideas. Differentiation simply means division , or ratio, or the ”rate of change” of one variable versus another. Integration simply means multiplication or a summation of a bunch of multiplications.
-
- Let’s start with the speedometer in your car. Miles per hour is simply the derivative of distance over time. velocity = ds / dt. Where “ds” is the change in distance, length, or space. Where “dt” refers to the change in time. Velocity is the rate of change of distance over time, v = ds / dt
-
- We are using velocity as an example but derivatives apply to any time you have the rate of change of one variable with respect to another variable. Always think of a derivative as a triangle. If one variable ( distance ) is plotted against another variable (time) the differential is the ratio of distance over time. It is the slope of the line. It is rise over run, or the hypotenuse of the right triangle with distance on the y-axis and time on the x-axis
-
- Velocity = ds /dt. Your speedometer is making this calculation continuously as you drive down the highway.
-
- Always think of “integrals as rectangles“, not triangles. Derivatives are division, Integrals are multiplication. Integrals are the area of a rectangle. To get the area of a rectangle you multiply the height versus the width. Derivatives and Integrals are opposites in a way.
-
- Acceleration is the derivative of velocity with time. a = dv / dt. Velocity is the integral of acceleration with time. v = Integral a * dt. “a” is acceleration in meters / second / second. ‘a“ = m/sec^2, “v” = meters / sec^2 * sec = meters / sec.
-
- If acceleration is plotted versus time. Acceleration on the y-axis and time on the x-axis, then, velocity is the area under the curve. If the acceleration is constant then the area is easy to find, it is length times width of the rectangle.
-
- If the acceleration is increasing at a constant rate then you have a line sloping upward and the area is that of a triangle. Velocity is ½ height * base of the triangle. If the acceleration is constantly changing, like going around a race track, then the velocity is the area under the acceleration curve. Here is where Integration becomes really powerful.
-
- You can approximate the area under any curve by making little rectangles, calculating the area of each one and adding them all up to get the total area under the curve. Whenever you have tedious, repetitive additions think Integration.
-
- Let’s illustrate with some examples: Suppose a belt was put around the equator of the Earth. Snug tight at a length of 25,000 miles. Suppose you could increase the length of the belt by your height so the belt hovered above the ground. Could you crawl underneath it?
-
------------------------- Circumference = 2*pi*radius
-
- Do you wonder how we get the differential and integral equations. It is easy. You simply look them up in a Standard Mathematical Tables. I use the C.R.C., Chemical Rubber company book published in 1931. The equations don’t change
-
------------------------- The derivative of 2*p*r = 2*pi = 6.28.
-
- Derivative means the rate of change of circumference with every increase in radius is 6.28 times. So, if you increase the radius by 1 foot the circumference increases by 6.28 feet. That is your height, 6.28 feet. The belt is 1 foot off the ground. Most people would guess the height of the belt to be a fraction of an inch. Do the math. This illustrates the power of being able to think in derivatives.
-
-------------- In Calculus talk, where “C“ is the circumference and “r“ is the radius:
-
---------------------------- dC( r ) /dr = a constant. d (2*pi*r) = 2*pi
-
- You drop a rock off a bridge and it takes 6 seconds to hit the water. How high is the bridge?
-
- Gravity has a constant acceleration on all objects, large and small. The constant acceleration of gravity at Earth’s surface is 32 feet per second per second. Acceleration,
-
-------------------------- a (t) = 32 ft/sec^2.
-
Velocity is the integral of acceleration with time. v = Integral a * dt.
-
------------------------- v = Integral 32 * dt
-
------------------------- v = 16 t^2
-
------------------------- note that the derivative of 16 t^2 /dt is 32 * dt
-
------------------------- the derivative of x^2 is 2x.
-
If the rock is falling with a velocity of 16*t^2 in 6 seconds it will have traveled 16*(6)^2 = 16 * 36 = 576 feet. The bridge is 576 feet above the water.
-
- Let’s suppose Linda owns a dress shop. She makes a profit of $100 a day on average. Gil being retired can invest her money and make an average of 5% over time. How much money will Gil have accumulated by the time Linda retires in 20 years?
-
- Linda works 365 days a year times $100 per day for 20 years = $730,000 total dollars. But, that does not include Gil’s investments. Gil invests the $36,500 the first year and that investment lasts at 5% for 19 years.
-
--------------------- 36,500 (1.05)^19 = $92,234
-
--------------------- (38,325 + 36,500) (1.05)^18 = $180,075
-
--------------------- ( 74,825 +36,500) (1.05) ^17 = $255,159
-
---------------------
---------------------
--------------------- and so on for 19 years, this becomes a long tedious summation of multiplications. It is better to recognize that long summations are best handled with “integration“. Recognize the compound interest is logarithmic curve. The formula for a logarithmic curve is e^x. “e” is the natural logarithm to the base = 2.718.
-
--------------------- The formula for the logarithmic curve we are developing :
-
-------------------- Total Dollars = Integral 36,500 dt *e^-.05(20-t)
-
-------------------- dt is one year increments.
-
-------------------- (20 - t ) means the each year you are compounding interest one less year out of the 20 years.
-
-------------------- $36,500 is what Linda makes in profit each year.
-
-------------------- The Total Dollars is the area under the logarithmic curve. We calculate the area by adding up (summations) all the individual rectangle areas calculated by multiplying e^-.05(20-t) * dt.
-
-------------------- Integral 36,500*dt * Integral e^-.05(20-t) from 0 to 20 years
-
------------------- 36,500 Integral ( e^(-1 + .05t))
-
-------------------- 36,500 ( e^.05*20 / .05 - e^.05 * 0 /.05)
-
-------------------- 36,500 ( e^1 - e^0 / .05) = 36,500 (2.718 - 1)/.05
-
-------------------- 36,500 ( 1.718 / .05 )
-
-------------------- 36,500 (34.37)
-
-------------------- $1,254,345.73
-
- Gil was so excited about his $1.2 million nest egg from Linda’s store that he decided to invest $100,000 in store improvements. However, he wanted to be sure Linda was maximizing profits with his investment.
-
- The average cost of the dresses that Linda bought on the used market was $30 per dress. After many years of experience Linda estimated her total market even if she gave the dresses away was 20,000. And, if she priced the dresses too high, say $400 each, she would not sell any. The best price was somewhere in between. The number sold, Q, would be 20,000 - 50 * p. Where “p” is the price of each dress. “Q” = quantity sold.
-
------------------------ Q = 20,000 - 50*p
-
------------------------ Cost = 100,000 - 30 * Q
-
------------------------ The rate of change of cost with respect to the quantity sold is the differential of the Cost = 30.
-
------------------------ The Revenue with respect to the quantity sold. Revenue = Quantity sold * price. Price = 400 - Q/50.
-
------------------------ Revenue (Q) = ( 400 - Q/50 ) * Q = 400 Q - Q^2/ 50
-
------------------------ The differential is the slope of the Revenue, dR/dQ = 400 - Q/25
-
------------------------ The differential of the Cost is the slope of the cost curve = 30.
-
------------------------ When the slope of the Cost curve = the slope of the Revenue curve the profit is maximum. This takes some thought.
-
- For example, if the slope of the Cost curve is greater than the slope of the Revenue curve it costs more to sell one more dress then the increase in revenue you get from selling one more dress. The same logic applies to the other side of the curve where the slope of the cost is less than the slope of the Revenue curve. So, we set the two slopes equal to each other in order to find the optimum quantity to sell for maximum profit:
-
------------------------ 400 - Q / 50 = 30
-
------------------------ Q = 9,250 dresses sold is the optimum quantity
-
------------------------ p = 400 - 9,250 / 50 = $215 price for each dress will maximize the profit from the business.
-
- Linda’s business is doing great. Gil’s investments are doing great. But, Linda is worried that Gil is spending too much time on the couch reading investment papers. He is not getting enough exercise. She takes his pulse on the couch and it is 75 beats per minute.
-
- Linda realizes that Gil has only so many heartbeats in his lifetime. When he gets to the last one he’s done. If Gil would only exercise he would lower his resting heart rate. Of course, when he exercise his heartbeat hastens. There is a trade off here. What is the optimum number of times you should exercise to maximize the number of heartbeats left in your life?
-
- How many additional heartbeats would Gil get during a week of exercise? Linda put Gil on the treadmill and got his heartbeat from 70 up to 130 during the exercise. She calculated that the exercise consumed 3,600 heartbeats.
-
------------------------- Gil’s resting heartbeat = 25/(Ex +1) + 50
-
------------------------- When exercise was zero his heartbeat was 75.
-
------------------------- With exercise his heartbeat was 62.5
-
------------------------- Ex is the number of exercises
-
------------------------- There are 10,000 minutes in a week.
-
------------------------ Total heartbeats in a week = 10,000 (25/(Ex+1) + 50) + 3,600* Ex
-
--------------------- Take the differential to find the slope of this curve:
-
--------------------- 250,000 + 1/(Ex+1)^2 + 3600 = 0
-
--------------------- Set this slope equal to zero to find the minimum point on the curve for Total heartbeats and solve for Ex. When a curve passes through a maximum or a minimum the slope always passes through zero.
-
---------------------- Ex = 7.5 exercises per week. Now Linda has Gil on the treadmill once a day.
-
- If Linda cut back her hours and worked 5 days a week for 50 weeks. And, Gil made 8% interest on her money. Their retirement nest egg would be $2,470,645. Gil would have made more money then Linda by lying on the couch. Of course he was using her money.
-
-------------------- 50,000 Integral e^.08t - 1 from 0 to 20 years
-
-------------------- 50,000 ( e^.08*20 / .08 - e^.08 * 0 /.08)
-
-------------------- 50,000 ( e^1.6 - e^0 / .08)
-
-------------------- 50,000 ( 4.95 / .08 - 12.5)
-
-------------------- 50,000 (61.9 -12.5)
-
-------------------- $2,470,645
-
- Hope you re not afraid to try a little calculus. It is very powerful math.
-
- November 30, 2019 2514 737
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----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Sunday, December 1, 2019 --------------------
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