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Exponential models Standard form(s) of an exponential • y=abx a is the initial value (y-intercept) b is the growth factor • y=aerx a is the initial value (y-intercept) r is the “per-capita rate” also called “exponential growth rate” • Conversion b=er, r=ln(b) When to use an exponential model • Things making more things – People making more people, bacteria making more bacteria, money making more money (interest) • Lots of identical events happening at random times – Radioactive decay (atoms decay at random times), Heat transfer (atoms bump into each other randomly) Bacteria • A colony of bacteria weighs 1mg at 12:00 and 3mg at 2:00. How much will it weigh at 6:00? • t=# of hours since 12:00 • n=# of mg of bacteria • (t,n) points are (0,1) and (2,3) • Bacteria make more bacteria, so this is an exponential model. n=abt • When t=0, Initial value (n) is 1. n=(1)bt • When t=2, n=3. 3=1b(2) • b=√3 n=(1)(√3) t • At 6 hours, I have n=(1)(√3)6 • 27mg of bacteria Standard form(s) of an exponential • y=abx a is the initial value (y-intercept) b is the growth factor • y=aerx a is the initial value (y-intercept) r is the “per-capita rate” also called “exponential growth rate” • Conversion b=er, r=ln(b) What is a percent rate? • Imagine that you have an investment earning 3% per year. • Per-cent means “for each one hundred.” • 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” • An investment of $100 earns 3 dollars per year. – An investment of $200 earns 6 dollars per year, – An investment of $300 earns 9 dollars per year, etc. What is compounding? • Now imagine that you have $100 in your account. After a year, you have more than $103 dollars! Why is that? • Well those $3 per year weren’t paid at the end of the year, they were paid in installments. • Each time you were paid money, your had more money in your account, so your rate went up • 3% of 100.25 is more than 3% of 100. So even after a month, when you have earned a quarter you are now earning money faster • This is because not just your $100 are earning money, but also the children of your $100 (that 25¢)are earning money. And the children of the 25¢ (the grandchildren of your $100 will also earn money for you.) What is APR? • Because of compounding, the percent earned on a 3%/year investment after a year is actually more than 3% of what you started with. • The percent you actually earn after a year is called your “annual percent yield” (APY) and it is higher than your “annual percentage rate” (APR) • Credit cards always tell you APR because it is always the smaller number. It’s not what you really pay (APY). What is continuous compounding? • Now imagine that you have $100 in your account. After a year, you have more than $103 dollars! Why is that? • Well those $3 per year weren’t paid at the end of the year, they were paid in installments. • However, in magical happyland (not the real world), you are not paid in installments, you are paid all the time. – Imagine walking 3 miles per hour. You don’t just wait an hour and then teleport three miles, you’re gaining distance all the time, as you’re walking. – Continuous compounding is the same idea: You are gaining money all the time, as you are waiting. • This means that every little tiny bit of money you have is earning 3% per year all the time. Every day, every hour, every minute you earn new money and calculate your 3% again. Compound interest formula nt æ rö A = P ç1+ ÷ è nø Compounding continuously ¥t æ rö A = P ç1+ ÷ è ¥ø We don’t know how to do this…. Yet. Compounding continuously Using a trick from calculus ¥t æ rö A = P ç1+ ÷ è ¥ø ¥ æ rö r ç1+ ÷ = e è ¥ø A = Pe rt Transitioning to per-capita rate • Imagine that you have an investment earning 3% per year. • Per-cent means “for each one hundred.” • 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” • But the number that actually goes in the equation is 3/100 • When compounding continuously, the math is easier if we talk about groups of 1 instead of groups of 100. Transitioning to per-capita rate • Imagine that you have an investment earning 3% per year. • 3% per year means “for each group of 100 dollars you have in your account, you earn 3 dollars per year.” • 3% also means “for each dollar I have, I earn 0.03 dollars per year.” • 1 dollar earns 3 cents per year – 10 dollars earns 30 cents per year – 100 dollars earns 3 dollars per year What is a per-capita rate? • Imagine that you have an investment earning 3% per year. • 3% also means “for each dollar I have, I earn 0.03 dollars per year.” • The per-capita rate is 0.03 dollars per year per dollar. • Sometimes people cancel the dollars and write this as 0.03/year • When compounding continuously, this is the number that goes in the r part of y=aerx Bacteria • A colony of bacteria has been measured as growing at an exponential rate of 0.033 bacteria per bacterium per hour. At 6 hours, there were 4.266 mg of bacteria. How much did the colony weigh at 0 hours? • t=# of hours • n=number of mg of bacteria • Exponential rate means use n=aert • Looking for the initial value… find a. • 4.266=ae0.033(6) • a≈3.5 General Strategy • Define your variables (x,y) • Write down your equation y=abx or y=aert • Fill in the parts you know – Values of a, b, or r – Values of x and y at specific times • Solve for the parts you don’t know – A missing value of a, b, or r. – Value of x or y at a different time. Suppose that at the start of the century, there are 25 teachers in a certain small town. Assuming the number of teachers grows exponentially with an exponential growth rate of .005/yr, what is the exponential growth model for this situation? a) b) c) d) e) P(t)=.005e25t P(t)=e25t P(t)=25e.005t P(t)=25et None of the above Suppose that at the start of the century, there are 25 teachers in a certain small town. Assuming the number of teachers grows exponentially with an exponential growth rate of .005/yr, what is the exponential growth model for this situation? t=# of years P(t)=# of teachers at time t P(t)=aert a=initial number of teachers =25. r=per-capita rate = 0.005 P(t)=25e0.005t C Doubling time problems • A colony of bacteria has been measured as growing at an exponential rate of 0.033 bacteria per bacterium per hour. How long will it take my colony to double in size? • t=# of hours • n=number of mg of bacteria • Exponential rate means use n=aert • Looking for a time…. Find t. • I started with a, when my colony doubled, I had 2a. • 2a=ae0.033t • 2=e0.033t • Ln(2)=0.033t • t=ln(2)/0.033 or approx 21 hours. How long will take an initial investment of $P to double in value under continuous compounding if the interest rate is 6%? a) b) c) d) e) About 4 years About 9 years About 12 years About 25 years About 42 years How long will take an initial investment of $P to double in value under continuous compounding if the interest rate is 6%? • • • • • • • • • • t=number of years A=amount after t years Continuous compounding, used A=Pert “How long” means find t. I started with P, when I doubled I had 2P. A=2P. 2P=Pe0.06t 2=e0.06t ln(2)=0.06t t=ln(2)/0.06 approx 11.6 years C about 12 years. Radioactive decay • Carbon-14 has a half life of about 5750 years. What is the corresponding exponential rate? • t=# of years • m=# of grams of carbon-14 • m=aert • Find r. • After 5750 years, I have half as much carbon-14. I started with a, so I have 0.5a left. • 0.5a=aer5750 • 0.5=er5750 • ln(0.5)=5750r • r=ln(0.5)/5750 or approx -0.00012 atoms per year per atom. • Bonus: the equation for carbon-14 decay is m=ae-0.00012t The half life of Cesium-137 is 30 years. What is the exponential decay rate, r, of this element? a) b) c) d) e) r = -.0552 r = -.0231 r = -.0431 r = -.0131 None of the above The half life of Cesium-137 is 30 years. What is the exponential decay rate, r, of this element? t=#of years m=#of grams of Cesium-137 m=aert Find r. I started with a. after 30 years (t=30), I had half as much (0.5a) 0.5a=aer30 0.5=er30 ln(0.5)=30r r=ln(0.5)/30 or approx -0.0231 atoms per year per atom B