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Lesson 4.2.1 Exponential Models & Logarithms
BRAINSTORMING ALGEBRA & STATISTICS
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INTRODUCTION The Cell Cycle The cell cycle is the process by which living cells replicate. The cell cycle is a relatively constant time period during which cells reproduce. For most mammals, the average cell cycle is very nearly one day. At the end of this cycle, a single cell experiences cytokinesis, where it divides into two cells. At the end of another cycle, each of these cells divides, and then there are four cells. In the discussion that follows, we assume that a population of cells grows from a single cell (beginning on day 0). This population will have adequate space and nutritional resources to allow continuous growth, and that no cells cease replication through a given time period. We begin by choosing a cell which has a cycle of 1 day. 1. Use a calculator with an exponent button ([xy] or [^]) to complete the table below. t A (Days) (Number of cells) 0 1 1 2 2 3 4 5 6 7 8 ⋮ ⋮ 15 19 ⋮ ⋮ t →
Rewrite A as a power of 2. 1 = 20 2 = 21 ⋮ ⋮
Figure 1: The cell cycle. From The Cell Cycle: Principles of Control by David O. Morgan. ©1999 – 2007, New Science Press. Used by permission.
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Lesson 4.2.1 Exponential Models & Logarithms 2. Write an exponential equation that gives the number of cells, A, after t days.
A = 3. Use trial and error on your calculator to estimate the time when there will be 2048 cells. In doing so, you are solving the equation 2t = 2048. Keep in mind that the equation which you have created is only a model, a function which estimates the number of cells after t days. It will not be perfectly correct, because all cell cannot be expected to complete their cycles in exactly one day, nor all at the same instant. In fact, over time, we can expect the cell divisions to occur at any time of the day, as their cycles lose synchronization. In fact, it is reasonable to expect that, while the experiment proceeds, any positive (whole) number of cells could be observed (up to some maximum number). When t = 7.65 days, we can expect around A = 27.65 200 cells. 4. Use trial and error with the exponential function on your calculator, to find a fractional number of days (not a whole number) when there are around 300 cells. In doing so, you are solving the equation 2t = 300. t = 5. Use trial and error with the exponential function on your calculator, to find a fractional number of days (not a whole number) when there are around 950 cells. In doing so, you are solving the equation 2t = 950. t =
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Lesson 4.2.1 Exponential Models & Logarithms t
6. Graph your exponential function, A = 2 , below. Use the points computed in the table from Question 1. A
260 240 220 200 180 160 140 120 100 80 60 40 20 0
0
1
2
3
4
5
6
7
8
t
7. Use your graph to estimate the (fractional) number of days when there will be 100 cells. NEXT STEPS Finding the exponent that causes an exponential function to yield a particular value can be challenging. This is particularly true when the exponent is not a whole number. Scientific calculators have a tool which can give the exponent of a base that will yield a particular value. That tool is called a logarithm. In problem 7 above you used a graph to determine the power of 2 which will yield 100. There are multiple ways to use a logarithm to give this answer. One way is to use a natural logarithm (denoted ln) function as follows (we will talk more about this later): ln 100 . ln 2 The 100 in the expression is the resulting number of cells we were trying to achieve, the 2 is the base of the exponential function.
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Lesson 4.2.1 Exponential Models & Logarithms TRY THESE 8. Use your calculator to evaluate this expression. ln 100 ln 2 9. How does your answer compare to the value determined graphically in problem 7? NEXT STEPS What is a Logarithm? A logarithm is used to give the value of an exponent in an exponential function. The simplest form of an exponential function looks like this: bx. In this function, the value, b, is the base of the exponential, and x is the exponent. The logarithm which can tell us the exponent in a base b exponential is a base b logarithm. Logarithms appear in various forms, but a base b logarithmic function looks like this: logb(x) This logarithm gives the power to which b must be raised to yield x. That is, logb(x) = the power to which b must be raised to yield x. For example, to evaluate log2(32), we ask “2 to the what yields 32?” Since 2 32, we know the answer is 5. log2(32) = 5 Evaluating the logarithm, log2(32) is equivalent to solving the equation 2t = 32. In summary, to evaluate logb(x), we ask the equivalent question, “b to the what yields x?”
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Lesson 4.2.1 Exponential Models & Logarithms TRY THESE 10. Give questions and equations that are equivalent to the logarithms below. Finally, solve the equations. Logarithm Equivalent Question Equation Solution log2(8)
2 to the what yields 8?
2t = 8
3
log3(81) log5(5) log7(49) 11. Exponential equations have a variable in an exponent. We often solve them using logarithms. In the table below, convert the equation to a question, then a logarithm, and solve it. Equation Equivalent Question Logarithm Solution 3t = 9
3 to the what yields 9?
log3(9)
2
10t = 1000 20t = 20 3t = 243 NEXT STEPS There are many types of logarithms, but your calculator only has a few. To evaluate logarithms, we use a mathematical trick to convert them to the natural logarithm like this: ln log . ln
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Lesson 4.2.1 Exponential Models & Logarithms TRY THESE 12. Use your calculator to evaluate the logarithms below. Equation Logarithm Convert to natural logarithm
Evaluate
4t = 9
log4(9)
1.585
7x = 135 9x = 1000 5x = 252 SUMMARY Let’s summarize what we know about logarithms. I. To evaluate the logarithm logb(x) ask “b to the what yields x?” II. The equation bt = x asks the same question: “b to the what yields x?” It is answered by the logarithm: t = logb(x). III. To evaluate a logarithm on a calculator, we use a trick to convert to the natural logarithm (ln). ln log ln TRY THESE Refer back to the exponential model which you used to quantify the number of cells in the population described in Problem 2. Use logarithms to determine when there will be approximately 3500 cells. 13. Rewrite the problem as an equation, using the model from 2, where the number of cells is A = 3500. Next, rewrite the equation as a base 2 logarithm, then evaluate on your calculator using the natural logarithm function.
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Lesson 4.2.1 Exponential Models & Logarithms TAKE IT HOME Before answering the questions below, review the following rule of exponents: b0 = 1 for any number a, a≠0. For example: 20 = 1, and also 150 = 1.
1
Rewrite each of the following logarithmic expressions as an exponential equation and then solve. (See #10 above). e. log5(125) = a. log2(4) = b. log3(27) =
f.
log3(3) =
c. log6(6) =
g. log5(1) =
d. log2(32) = 2
Rewrite each of the following exponential equations as a logarithmic expression and then solve for t. (See Question 11 in the classroom activity above.) e. 3t = 1 a. 3t = 81
b. 2t = 16
f.
4t = 64
c. 5t = 625
g. 5t = 5
d. 2t =128
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Lesson 4.2.1 Exponential Models & Logarithms 3
Write the following exponential equations as a logarithmic expression and use your calculator to evaluate your answer. (See Question 12 in the classroom activity above.) a. 2x = 7 e. 3t = 300
b. 5n = 100
f.
4w = 190
c. 3z = 65
g. 9r = 250
d. 2t = 70
4
The exponential equation below is used to model the number of people who live in Washmont City, where A represents the number of people who live in Washmont after t years. A = 3t
a. Estimate the number of people who will be living in Washmont after 10 years? (What is the value of A when t = 10?) Round your answer to the nearest whole person.
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Lesson 4.2.1 Exponential Models & Logarithms b. Estimate the number of years required for the population of Washmont to reach 2000 people? (What is the value of t when A = 2000?) Round your answer to the nearest hundredth.
5
The exponential equation below is used to model the population of bacteria in a culture, where A represents the number of bacteria present after t days. A = 5.4t a. Estimate the number of bacteria that will be present after 4 days? (What is the value of A when t = 4?) b. Estimate the number of days for 300 bacteria to be present? (What is the value of t when A = 300?) Round your answer to the nearest hundredth.
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Lesson 4.2.2 Exponential Models & Logarithms
BRAINSTORMING ALGEBRA & STATISTICS
STUDENT NAME
DATE
INTRODUCTION Compound Interest When you invest money in a fixed‐rate interest earning account, you receive interest at regular time intervals. The first time you receive interest, you receive a percentage of your initial investment (the principal). After that, you continue to receive interest from your principal, but your previous interest also earns interest. This excellent phenomenon is known as compound interest. Suppose that you have a fixed‐rate interest earning account which earns 8% interest at the end of every year, and that you have deposited $10,000. Let’s call the time of the initial investment t = 0. 1. How much money will be in the account after one year when t = 1? (Remember to count principal plus interest!) 2. What percent of the original principal will we earn as interest? 3. What percent of the original principal do we get to keep? 4. Adding these percentages together, what is the total percentage of the principal that becomes the new balance? 5. How does this percentage appear in decimal form, if we wish to use it for computing account balances?
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Lesson 4.2.2 Exponential Models & Logarithms The value you determined in step 5 is a multiplier for determining a given year’s balance. If we multiply the principal by this value 3 times, we would get the balance on the third year. Fill in the balances for the given years in the table below. The year 2 balance is provided. See if you can get the same value. t (Years) 0 1 2 3 4 5
A (Account Balance) 10000 11664
6. Maybe you determined the year 4 balance by multiplying by the interest multiplier four times. How could you do this more efficiently by using the exponent button on your calculator? 7. Describe a method for determining the account balance in 25 years, using mathematical exponents. 8. Compute the balance for the years in the table below. Use the exponent function on your calculator. t (Years) 15 25
A (Account Balance)
9. Write an equation that gives the account balance, A, after t years. The equation should use exponents. A =
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Lesson 4.2.2 Exponential Models & Logarithms 10. Graph your exponential function below. Note that you have already computed the balance for several years above. Compute a few additional points to help complete the picture.
A
70000 60000 50000 40000 30000 20000 10000 0
0
5
10
15
20
25
t
11. Use your graph to estimate the time when the account will contain $40000. NEXT STEPS If all went as planned, you discovered a formula for computing compound interest, when interest is earned once each year. If P is the principal, the initial deposit, of some investment into a fixed‐rate interest earning account, and r is the annual interest rate, then the multiplier for determining the next year’s balance is 1 + r (keep 100% of the original investment and add in a proportion of r for interest. Remember, the multiplier needs to be in decimal, not percentage, form). The balance at year t = 1 is A = P(1 + r). The balance at year t = 2 is A = P(1 + r)2. At year t, the balance is A = P(1 + r)t This is an exponential function, because the only changing quantity is the variable t in the exponent. The multiplier, 1 + r, is the base of the exponent.
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Lesson 4.2.2 Exponential Models & Logarithms TRY THESE 12. Suppose you invest $5000 at 12% annual interest in a fixed‐rate interest earning account, where interest is compounded once each year. What will the account balance be in 10 years? 13. What will the account balance be in 30 years? NEXT STEPS In reality, interest earning accounts receive interest more than once a year. Suppose an account earns 8% interest, compounded quarterly (four times a year). When this occurs, the 8% is divided by four, and you only receive 2% at each compounding. This is actually better because it gives your interest more times to compound, and in the end you earn slightly more than 8% annually. If the investment is for 10 years, you would actually receive interest 4⋅10 = 40 times. In general, when the annual interest rate is r, and interest is compounded n times each year, the actual rate is r/n, so the multiplier for computing interest is 1 + r/n. Interest is earned n⋅t times, so the exponent for the multiplier is n⋅t. These ideas give a new compound interest formula, ⁄ 1 . In this equation, A is account balance, P is principal, r is annual interest rate, t is time in years, and n is the number of times interest is compounded each year. TRY THESE 14. Suppose you invest $2500 at 10%, compounded quarterly, in a fixed‐rate interest earning account. How much will you have in 17 years? 15. How much will you have in 25 years? © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.2.2 Exponential Models & Logarithms NEXT STEPS In our last lesson, we learned how to use logarithms to solve for an exponent in an equation. We learned that logb(x) = the power to which b must be raised to yield x, and that, to evaluate logb(x), we simply ask “b to the what yields x?”. For example, to evaluate log6(216), we ask, “6 raised to the what yields 216?”. Since 63 = 216, the answer is 3. log6(216) = 3 TRY THESE Evaluate the following logarithms without a calculator. 16. log2(64) = 17. log5(125) = 18. log9(81) = 19. log4(47) = 20. log6(615) = 21. logb(bx) = © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.2.2 Exponential Models & Logarithms NEXT STEPS Also in our last lesson, we learned to evaluate logarithms with a calculator. We learned that calculators have a special logarithm function called the natural logarithm, which is denoted as ln. With the natural logarithm, we can evaluate any logarithm as follows. ln log . ln TRY THESE Evaluate the following logarithms with a calculator. 22. log6(100) = 23. log8(475) = 24. log10.5(1298) = 25. log7(7123) = 26. log12(1226) = NEXT STEPS If you paid attention during the two previous problem sets, you may have discovered a principal in logarithms. If you did not see it, check again. Look at problems 19, 20, 24, 25, 26, but look particularly at problem 21, which says it all. If this was done correctly it tells us that logb(bu) = u. This key idea is really the main purpose of logarithms. It tells us that if a logarithm is applied to an exponential of the same base, it gives us the exponent. The logarithm is the inverse operation of the exponential.
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Lesson 4.2.2 Exponential Models & Logarithms Watch how we use this idea to solve the following equation. 1000 ⋅ 1.10 20000 Here is our given equation. 1000 ⋅ 1.10 20000 1000 ⋅ 1.10 1000
1.10
20
log
log
.
1.10 2
log
.
20000 1000
We need to isolate the exponential, so we divide by 1000. The exponential is isolated. Its base is 1.10.
.
20
20
Here we apply a logarithm of the same base to each side of the equation. Now we use the rule that logb(bu) = u.
2
ln 20 ln 1.1
Convert to a natural logarithm to evaluate on a calculator.
2
31.4314
2 2
31.4314 2 15.7157
We have isolated 2x, but now we must divide by 2 to finish. Divide by 2. Here is the final answer.
We can verify this final answer by substituting into the original equation: 20000.002 1000 ⋅ 1.10 . The solution found above is quite close to being exact. In summary, to solve an exponential equation, I. Isolate the exponential (the base which is raised to a power). II. Apply a logarithm of the same base as the exponential to each side. III. Use the rule logb(bu) = u to simplify the equation. IV. Evaluate the resulting expression with a calculator using the rule logb(x) = ln(x)/ln(b).
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Lesson 4.2.2 Exponential Models & Logarithms TRY THESE Use the steps above to solve the equations below. 27. 3⋅1.23x = 93 28. 36⋅0.952x = 18 Recall that the balance of a fixed‐rate interest earning account is given by the formula: 1
.
In this equation, A is account balance, P is principal, r is annual interest rate, t is time in years, and n is the number of times interest is compounded each year. 29. Suppose you invest $12000 in a fixed‐rate interest earning account, at 9% interest, compounded n = 4 times each year. Substitute the values given into the equation for compound interest above.
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Lesson 4.2.2 Exponential Models & Logarithms 30. When will the account contain $20000? Solve your equation for the variable, t. 31. Suppose the account earned 12% interest, compounded twice a year. When would it contain $20000?
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Lesson 4.2.2 Exponential Models & Logarithms TAKE IT HOME 1
2
Art puts $15500 in a bank account earning 6% interest. If the interest is compounded monthly (12 times a year) how much money will Art have in his account after 3 years? Ronda has $2300 in an investment earning 4% interest. If the interest is compounded semi‐annually (two times a year), answer the following: a. For each value of t given in the table, find the amount of money that Ronda will have in her account after that many years. Record your answers in the space provided in the table below. Value of t A 5 10 15 20 25 30 b. Use the values above to graph a curve that models the amount of money Ronda has in her account after t years. A
8000
7000
6000
5000
4000
3000
2000
1000
0
0
5
10
15
20
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25
30
35
t
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Lesson 4.2.2 Exponential Models & Logarithms
3
4
c. Use the graph in part (b) to approximate how many years it will take Ronda to have $3500 in her account. d. Use logarithms to find the number of years it will take Ronda to have $3500 in her account. (See problems 29‐31 in the lesson above) Scott would like to buy a car in 4 years. He needs $25000 to buy a new car. Currently Scott has $19000 and he has decided to place the money in a bank account and earn interest in hopes that he will be able to buy his new car in 4 years. If he places the money in an account that earns 4% interest that compounds annually, will he reach his goal of $25000 in four years? Evaluate the following logarithms without the use of a calculator. (See #16‐21 above) a. log3(3) = d. log7(712) = e. log5(59) =
b. log2(128) = c. log4(64) = 5
Evaluate the following logarithms using a calculator. (See #22‐26 above) d. log4(434) = a. log2(91) = e. log2(214) =
b. log3(19) = c. log7(213) =
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Lesson 4.2.2 Exponential Models & Logarithms 6
Solve the following equations. (See Questions 27 & 28 in the above classroom activity.) a. 63.42t = 87 c. 40.322t = 3
b. 59.13x = 130
d. 70.974x = 4
For the following problems, refer to Questions 29 through 31 from the classroom activity in this handout. 7 If you placed $34000 in an account that earned 5% interest, compounded yearly, how much time would it take to have a balance of $45000? 8 Akira has $12000 to put in the bank. He found an account that earns 6% interest and is compounded quarterly. How long will Akira need to keep his money in that account if he wants to have a total of $30000?
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Lesson 4.2.2 Exponential Models & Logarithms 9
Paula needs $10000 for a down payment on a car; currently she has $5000. Paula is going to deposit her money in an account that earns 7% interest and compounds monthly. How long will Paula need to keep her money in the account?
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Lesson 4.3.1 The Natural Exponential and Logarithm
BRAINSTORMING ALGEBRA & STATISTICS
STUDENT NAME
DATE
INTRODUCTION Review In our previous lesson, we learned that fixed‐rate interest earning accounts grow according to the compound interest formula, 1
.
In this formula, A is account balance at time t, P is the principal (the initial investment), r is the annual interest rate, and n is the number of times that the account earns interest each year.
TRY THESE 1. Suppose you invest $15000 at 10% annual interest, compounded semi‐annually (n = 2). What will the account balance be in 10 years? Suppose you invest $15000 at 10% interest, compounded semi‐annually. You wish to know the time when your money has doubled (so the account balance is A = $30000). 2. Substitute all known values into the compound interest equation. There should be only one unknown, t. Simplify the expression. 3. Isolate the exponential by dividing each side by the principal.
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Lesson 4.3.1 The Natural Exponential and Logarithm 4. What is the base of the exponential? 5. Apply a logarithm of the same base to each side, and simplify using the fact that logb(bu) simplifies to u (in this formula, the base is b). 6. Change the logarithm in the equation to a natural logarithm using the rule: log
. Evaluate
the result on your calculator. 7. Solve the equation for time, t, when the account balance has doubled to $30000.
NEXT STEPS
Continuous Growth We have mentioned that the more times that interest is applied to a fixed‐rate interest earning account, the more money you will earn. This is because more compoundings means more times for your interest to earn interest. But, is there a limit to this? What if compound interest every hour? What about every minute or second? If we could compound interest every second, our account balance would nearly be growing continuously! To investigate whether our money growth is limited with increased compounding, let’s look at an extraordinarily unrealistic example to see where things go. You have found a fixed‐interest earning bank account that earns interest as many times a year as you would like, at an interest rate of 100%. The only catch is that the biggest principal you can invest is $1. The © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.1 The Natural Exponential and Logarithm question we need to answer is how many compoundings per year should we suggest in order to maximize your account balance. You invest a principal of P = $1. The interest rate is r = 100% = 1.00. You decide to invest the money for 1 year initially, so t = 1. We will experiment with different values of n to see how we can maximize the return on our investment. The compound interest formula becomes: 1
1
1
If we decide to compound interest once, then n = 1, and the balance at 1 year would be 1 2
1 1
That makes sense. After 1 year at 100% interest our $1 principal has grown to $2. Not a very enticing investment. But, if we decide to compound more frequently, what would that do to our investment? Let’s compound interest twice, where n = 2. At the end of year 1 our balance is 1 2.25
1 2
Incredible! Semi‐annual compounding has earned us an extra $0.25! Kaa‐ching! Let’s look deeper. Fill in the table for account balances corresponding to increasing values of n. n A=(1+1/n)n (compoundings) 1 2 100 10,000 1,000,000 10,000,000
2 2.25
8. Is your investment growing without limit? Should you look for other ways to get rich? 9. What is the limiting value that your investment is approaching? © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.1 The Natural Exponential and Logarithm NEXT STEPS If all went well, you learned that more compoundings per year does increase your earnings. Unfortunately, you also learned that there is a limit to those earnings. As the number of compoundings per year approaches infinity, your account balance approaches a number that mathematicians call e. e = 2.718281828459…
The Base of the Natural Exponential The number, e, is very special in mathematics. In fact, you have discovered that the mathematical expression, 1 1 approaches e as n increases without limit. In our example, it discovered that the balance of an account which earns interest almost continuously approaches e. In fact, it is not difficult to show, that if we do compound interest continuously, the account balance formula becomes . Here we see e as the base of an exponential function that models continuous growth. To compute an exponential, base e, on your calculator, look for the [ex] key.
TRY THESE 10. Suppose that you deposit $10000 at 7% interest into a fixed‐rate continuously compounded interest account. What is the formula for the account balance? 11. What will be the account balance in 5 years? 12. Use trial and error to find the time when your account balance has doubled from the principal investment (when A = $20000). © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.1 The Natural Exponential and Logarithm 13. Compute the account balance, A, corresponding to the times, t, in the table below. t A 0 $10000 6 12 18 24 14. Plot the continuously compounded interest function below.
A
70000 60000 50000 40000 30000 20000 10000 0
0
6
12
18
24
t
15. From the graph, visually estimate the time when the account balance is $20000.
NEXT STEPS We know that every exponential function, bx, there is a logarithm of the same base, logb(x), that returns the exponent: logb(bu) = u. Now that we have learned of the natural exponential function, ex, it is time to formally define its corresponding logarithm, the natural logarithm, loge(x). The natural logarithm is so commonly encountered that it has been given a shorthand notation: ln(x). ln(x) is shorthand for loge(x).
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Lesson 4.3.1 The Natural Exponential and Logarithm We have seen the natural logarithm several times, and have used it to evaluate other logarithms. For now, the most important fact about the natural logarithm is that it returns the exponent of the natural exponential. ln(eu) = u.
TRY THESE 16. Refer again to our $10000 investment into a fixed 7% interest earning account. Using the continuously compounded interest formula, solve for the time when the account balance is equal to $20000. First, divide by the principal, then apply the natural logarithm to each side. Use your calculator to evaluate. 17. Does your answer match (roughly) the estimates in Problems 12 and 15? 18. When will the account balance be $30000? Use the method from Problem 16 to answer. 19. Suppose the interest rate is 10%. When will the account balance reach $100000?
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Lesson 4.3.1 The Natural Exponential and Logarithm TAKE IT HOME There are two main formulas for calculating an account balance. The first one, shown below, is used when interest is compounded n times per year. 1
The second equation, shown below, is used when interest is compounded continuously. A = Pert For the following questions, make sure you are using the correct equation. Keep an eye out for key words, when you see continuously compounded you know you need to use the second equation. Also, if the problem tells you that interest is compounded quarterly or a certain number of times a year, then you are being given a value for n and should be using the first equation. 1. Samuel is investing $5000 in an account that has an interest rate of 8%. How much money will Samuel have in his account after 3 years if: a. The interest is compounded semi‐annually (2 times a year)?
b. The interest is compounded monthly (12 times a year)?
c. The interest is compounded weekly (52 times a year)?
d. The interest is compounded continuously?
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Lesson 4.3.1 The Natural Exponential and Logarithm 2. Bianca invests $10000 in an account that has an interest rate of 5%. The interest is compounded continuously. Use this information to answer the following questions. a. For each value of t given in the table, find the amount of money that Bianca will have in her account after that many years. Record your answers in the space provided in the table below.
Value of t 5 10 15 20 25
A
b. Use the values above to graph a curve that models the amount of money Bianca has in her account after t years. A 35000
30000
25000
20000
15000
10000
5000
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
t
c. Use the graph in part (b) to approximate the number of years it will take for Bianca to have $25000 in her account.
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Lesson 4.3.1 The Natural Exponential and Logarithm
d. Use logarithms to determine the number of years until Bianca has $25000 in her account. (See Questions 16 – 19 in the lesson above.) 3. Thomas has $7000 to invest in an account. He has to choose between two bank accounts. Account A has a 4.5% (in decimal form that is 0.045) interest rate and the interest is compounded continuously. Account B has a 5.5% interest rate and the interest is compounded quarterly (4 times a year). a. How much money will be in account A if the money was invested for 2 years? b. How much money will be in account B if the money was invested for 2 years? c. Which account should Thomas invest in, if he is going to keep the money in the account for 2 years?
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Lesson 4.3.1 The Natural Exponential and Logarithm 4. Juan invests $12000 in an account that has a 4.25% (in decimal form that is 0.0425) interest rate. The interest is compounded continuously. How many years will it take for Juan’s money to grow to $20000? 5. Samantha invested some money in an account that has a 6% interest rate. If the interest is compounded monthly, and after 5 years Samantha has $15500, how much money did she invest initially?
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Lesson 4.3.2 Exponential Growth and Decay
BRAINSTORMING ALGEBRA & STATISTICS
STUDENT NAME
DATE
INTRODUCTION Exponential Growth and Decay. temp
18 In our previous lesson, we experimented with the idea that a 16 fixed‐rate interest earning account could grow continuously. If 14 such an account actually existed, the account balance would be 12 given by 10 A = Pert. 8 6 In this formula, P is the starting balance, r is the annual interest 4 rate, and t is time in years. 2 In this lesson, we learn that any phenomenon that grows or 0 0 20 40 60 80 100 120 140 160 180 time decays exponentially can be modeled by a base e exponential model, just like the compound interest formula above. Figure 1: An exponential decay model. A generic model which applies to every exponential growth or decay application is A = A0 ekt. In this model, A represents an amount of something that grows or decays over time. The constants A0 and k are analogous to P and r in the compound interest formula. A0 is the initial amount (of something) at time t = 0, and k is the instantaneous growth/decay rate. In a growth model, k is positive, and in a decay model, k is negative.
Newton’s Law of Cooling As an object cools in a room whose temperature is held constant, its temperature decays exponentially as it approaches room temperature (this is Newton’s Law of Cooling). Let A represent the object’s temperature at time t, measured in degrees Celsius above room temperature (so A = 0 is room temperature). A0 represents the object’s initial temperature at time t = 0. The following facts were determined by fitting an exponential model to real temperature values as pictured in Figure 1: An exponential decay model.. A temperature probe measured an initial temperature of 18° F (so A0 = 18), above room temperature, and according to the model, it cooled at an instantaneous rate of 1.7% (so k = ‐0.017) per second. © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.2 Exponential Growth and Decay The exponential decay model for the temperature is A = 18 e‐0.017t.
TRY THESE 1. How many degrees above room temperature is the object in 30 seconds? Use the graph in Figure 1 to estimate the answer to this question. 2. Use the exponential decay formula above to answer Question 1 again. Do your answers agree (approximately)? Suppose we need to know when the probe will reach 2° above room temperature. 3. Use the graph in Figure 1 to estimate the time when the probe’s temperature is 2° above room temperature. 4. In the exponential decay formula, set the temperature, A, equal to 2° (degrees above room temperature), and solve for the time when this occurs. This is done by first dividing by the initial temperature (18°), applying the natural logarithm (ln) to each side, and using the fact that ln(eu) = u. 5. Does the time in Question 4 agree (approximately) with your estimate from Question 3?
NEXT STEPS When taking medications internally, the amount of medication in the body breaks down over time, decaying exponentially. For example, the pain reliever ibuprofen has an hourly decay rate of about 35% (0.35).
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Lesson 4.3.2 Exponential Growth and Decay TRY THESE Suppose a person takes an 800 mg tablet of ibuprofen. Use the fact that ibuprofen has an hourly decay rate of 35% to answer the following questions. 6. Give the formula which models the amount of ibuprofen in her body after t hours. 7. If a person takes an 800 mg tablet of ibuprofen, how much remains in her system after 2 hours? 8. Compute the amount of ibuprofen in her body for the times listed in the table below. A (Ibuprofen Amount) 800
t (Hours) 0 2 4 6 8 10
9. Graph the exponential decay model using the points you computed above. A
800 700 600 500 400 300 200 100 0
0
1
2
3
4
5
6
7
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t
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Lesson 4.3.2 Exponential Growth and Decay 10. Use the graph above to estimate the half‐life of ibuprofen (the time when only half remains). 11. Using a process similar to that used in Question 4, solve the formula from Question 6 for the half‐life of ibuprofen. Begin by setting the amount of ibuprofen, A, to half of its original amount. 12. Suppose the patient’s doctor does not want her ibuprofen level to drop below 150 mg. Additionally, it should never be above 1000 mg. Use the graph above to determine how frequently she should take the medication.
13. Using a process similar to that used in Question 4, solve the formula from Question 6 for the time when her ibuprofen level reaches 150 mg. Does this agree with your answer from Question 12?
NEXT STEPS Suppose that a population has initial size 1000 at time t = 0, and grows exponentially with a 20% (0.20) instantaneous annual growth rate.
TRY THESE 14. Give the formula which models the population size in t years. © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.2 Exponential Growth and Decay 15. Use the formula to estimate the population size in ten years. 16. Compute the amount of ibuprofen in her body for the times listed in the table below. A (Population Size) 1000
t (Hours) 0 2 4 6 8 10
17. Graph the exponential growth model using the points you computed above. A
8000 7000 6000 5000 4000 3000 2000 1000 0
0
1
2
3
4
5
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8
9
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t
18. Use your graph above to estimate the time when the population size is 3000. 19. Using the formula from Question 14, solve for the time when the population time is 3000. © 2013 GUTH, YOUNG, AND NITTA A BRAINSTORM OF ALGEBRA AND STATISTICS, VERSION 2.0, SACWAY™ ‐ STUDENT HANDOUT
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Lesson 4.3.2 Exponential Growth and Decay 20. Use your graph to estimate the population’s doubling time. 21. Using the formula from Question 14, solve for the population’s doubling time.
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Lesson 4.3.2 Exponential Growth and Decay TAKE IT HOME Previously, we examined the radioactivity levels in the Nevada Area 2 test site after 1985. We determined that an exponential model was an appropriate fit for the radiation level after t years. 2000 1800
Radiation (mR/yr)
1600 1400 1200 1000 800 600 400
Years 200
0
5
10
15
20
25
30
After analysis, it is learned that the exponential model predicts an initial radiation level of 2000.2 mR/yr, with an instantaneous decay rate of 5.3%. 1. Use the form, A = A0 ekt, to give the formula for the exponential decay model for the Area 2 radiation level after t years. 2. Use the graph to estimate the ½ life of the radiation at Area 2. 3. Solve the exponential decay model in Question 1 for the ½ life (by setting the radiation level, A, to half of its original amount).
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Lesson 4.3.2 Exponential Growth and Decay 4. Use the formula from Question 1 to estimate the time when the radiation level was 500 mR/yr. In February 2004, Facebook was born as a social networking tool. The company has been tracking its membership since December 2004. The graph below represents membership counts (in millions) at t months, beginning in December 2004 (through July 2010). An exponential model fits the data appropriately. 600
Users (millions)
500
400
300
200
100
Months 0
0
10
20
30
40
50
60
70
80
After analysis, it is learned that the exponential model predicts an initial membership of 3.32 million, and an instantaneous growth rate of 6.6%. 5. Use the form, A = A0 ekt, to give the formula for the exponential growth model for the number of Facebook members after t months. 6. Use the graph to estimate the doubling time of the Facebook membership. This might be easiest by estimating the time for the membership to grow from 100 to 200 million.
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Lesson 4.3.2 Exponential Growth and Decay 7. Solve the exponential growth model in Question 5 for the doubling time (by setting the membership, A, to twice of its original amount). 8. Use the formula from Question 5 to estimate the time when the membership was 400 million.
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