LESSON #25  EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II


 Bernadette Gibson
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1 1 LESSON #5  EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II Eponential functions are very important in modeling a variety of real world phenomena because certain things either increase or decrease by fied percentages over given units of time. You looked at this in Common Core Algebra I and in this lesson we will review much of what you saw. Eercise #1: Suppose that you deposit money into a savings account that receives 5% interest per year on the amount of money that is in the account for that year. Assume that you deposit $400 into the account initially. (a) How much will the savings account increase by over the course of the year? (b) How much money is in the account at the end of the year? (c) By what single number could you have multiplied the $400 by in order to calculate your answer in part (b)? (d) Using your answer from part (c), determine the amount of money in the account after and 10 years. Round all answers to the nearest cent when needed. (e) Give an equation for the amount in the savings account as a function of the number of years since the $400 was invested. (f) Using a table on your calculator determine, to the nearest year, how long it will take for the initial investment of $400 to double. Provide evidence to support your answer. The thinking process from Eercise #1 can be generalized to any situation where a quantity is increased by a fied percentage over a fied interval of time. This pattern is summarized below: INCREASING EXPONENTIAL MODELS If quantity Q is known to increase by a fied percentage p, in decimal form, then Q can be modeled by where represents the amount of Q present at and t represents time. Eercise #: Which of the following gives the savings S in an account if $50 was invested at an interest rate of 3% per year? (1) S 504 t t (3) S () S t (4) S t
2 Decreasing eponentials are developed in the same way, but have the percent subtracted, rather than added, to the base of 100%. Just remember, you are ultimately multiplying by the percent of the original that you will have after the time period elapses. Eercise #3: State the multiplier (base) you would need to multiply by in order to decrease a quantity by the given percent listed. (a) 10% (b) % (c) 5% (d) 0.5% DECREASING EXPONENTIAL MODELS If quantity Q is known to decrease by a fied percentage p, in decimal form, then Q can be modeled by where represents the amount of Q present at and t represents time. Eercise #4: If the population of a town is decreasing by 4% per year and started with 1,500 residents, which of the following is its projected population in 10 years? Show the eponential model you use to solve this problem. (1) 9,30 (3) 18,503 () 76 (4) 8,310 Eercise #5: The stock price of WindpowerInc is increasing at a rate of 4% per week. Its initial value was $0 per share. On the other hand, the stock price in GerbilEnergy is crashing (losing value) at a rate of 11% per week with an initial value of $10. (a) Model both stock prices using eponential functions. (b) Then, find when the stock prices will be equal graphically, to the nearest week. Draw a well labeled graph to justify your solution. y
3 3 APPLICATIONS EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II HOMEWORK 1. If $130 is invested in a savings account that earns 4% interest per year, which of the following is closest to the amount in the account at the end of 10 years? (1) $18 (3) $168 () $19 (4) $34. A population of 50 fruit flies is increasing at a rate of 6% per day. Which of the following is closest to the number of days it will take for the fruit fly population to double? (1) 18 (3) 1 () 6 (4) 8 3. If a radioactive substance is quickly decaying at a rate of 13% per hour approimately how much of a 00 pound sample remains after one day? (1) 7.1 pounds (3) 5.6 pounds ().3 pounds (4) 15.6 pounds 4. A population of llamas stranded on a dessert island is decreasing due to a food shortage by 6% per year. If the population of llamas started out at 350, how many are left on the island 10 years later? (1) 57 (3) 10 () 58 (4) Which of the following equations would model a population with an initial size of 65 that is growing at an annual rate of 8.5%? t P (3) P (1) t () P t (4) P 8.5t The acceleration of an object falling through the air will decrease at a rate of 15% per second due to air resistance. If the initial acceleration due to gravity is 9.8 meters per second per second, which of the following equations best models the acceleration t seconds after the object begins falling? (1) () a a t (3) a t t (4) a t
4 7. Red Hook has a population of 6,00 people and is growing at a rate of 8% per year. Rhinebeck has a population of 8,750 and is growing at a rate of 6% per year. In how many years, to the nearest year, will Red Hook have a greater population than Rhinebeck? Show the equation or inequality you are solving and solve it graphically A warm glass of water, initially at 10 degrees Fahrenheit, is placed in a refrigerator at 34 degrees Fahrenheit and its temperature is seen to decrease according to the eponential function (a) Verify that the temperature starts at 10 degrees Fahrenheit by evaluating T 0. h T h (b) Using your calculator, sketch a graph of T below for all values of h on the interval 0 h 4. Be sure to label your yais and y intercept. (c) After how many hours will the temperature be at 50 degrees Fahrenheit? State your answer to the nearest hundredth of an hour. Illustrate your answer on the graph your drew in (b). REASONING 9. Percents combine in strange ways that don't seem to make sense at first. It would seem that if a population grows by 5% per year for 10 years, then it should grow in total by 50% over a decade. But this isn't true. Start with a population of 100. If it grows at 5% per year for 10 years, what is its population after 10 years? What percent growth does this represent?
5 5 LESSON #6  MINDFUL MANIPULATION OF PERCENTS COMMON CORE ALGEBRA II Percents and phenomena that grow at a constant percent rate can be challenging, to say the least. This is due to the fact that, unlike linear phenomena, the growth rate indicates a constant multiplier effect instead of a constant additive effect (linear). Because constant percent growth is so common in everyday life (not to mention in science, business, and other fields), it's good to be able to mindfully manipulate percents. Eercise #1: A population of wombats is growing at a constant percent rate. If the population on January 1 st is 107 and a year later is 1079, what is its yearly percent growth rate to the nearest tenth of a percent? Eercise #: Now let's try to determine what the percent growth in wombat population will be over a decade of time. We will assume that the rounded percent increase found in Eercise #1 continues for the net decade. (a) After 10 years, what will we have multiplied the original population by, rounded to the nearest hundredth. Show the calculation. (b) Using your answer from (a), what is the decade percent growth rate, to the nearest tenth? Eercise #3: Let s stick with our wombats from Eercise #1. Assuming their growth rate is constant over time, what is their monthly growth rate to the nearest tenth of a percent? Assume a constant sized month. Eercise #4: If a population was growing at a constant rate of % every 5 years, then what is its percent growth rate over at year time span? Round to the nearest tenth of a percent. (a) First, give an epression that will calculate the single year (or yearly) percent growth rate based on the fact that the population grew % in 5 years. (b) Now use this epression to calculate the percent growth over years, to the nearest tenth.
6 Eercise #5: World oil reserves (the amount of oil unused in the ground) are depleting at a constant % per year. We would like to determine what the percent decline will be over the net 0 years based on this % yearly decline. (a) Write and evaluate an epression for what we would multiply the initial amount of oil by after 0 years. (b) Use your answer to (a) to determine the percent decline after 0 years. Be careful! Round to the nearest percent. 6 Eercise #6: A radioactive substance s halflife is the amount of time needed for half (or 50%) of the substance to decay. Let s say we have a radioactive substance with a halflife of 0 years. (a) What percent of the substance would be radioactive after 40 years, to the nearest tenth of a percent? (b) What percent of the substance would be radioactive after only 10 years? Round to the nearest tenth of a percent. (c) What percent of the substance would be radioactive after only 5 years? Round to the nearest tenth of a percent.
7 7 MINDFUL MANIPULATION OF PERCENTS COMMON CORE ALGEBRA II HOMEWORK APPLICATIONS 1. A quantity is growing at a constant 3% yearly rate. Which of the following would be its percent growth after 15 years? (1) 45% (3) 56% () 5% (4) 63%. If a credit card company charges 13.5% yearly interest, which of the following calculations would be used in the process of calculating the monthly interest rate? (1) () (3) (4) The county debt is growing at an annual rate of 3.5%. What percent rate is it growing at per years? Per 5 years? Per decade? Show the calculations that lead to each answer. Round each to the nearest tenth of a percent. 4. A population of llamas is growing at a constant yearly rate of 6%. At what rate is the llama population growing per month? Please assume all months are equally sized and that there are 1 of these per year. Round to the nearest tenth of a percent.
8 5. Shana is trying to increase the number of calories she burns by 5% per day. By what percent is she trying to increase per week? Round to the nearest tenth of a percent If a bank account doubles in size every 5 years, then by what percent does it grow after only 3 years? Round to the nearest tenth of a percent. Hint: First write an epression that would calculate its growth rate after a single year. 7. An object s speed decreases by 5% for each minute that it is slowing down. Which of the following is closest to the percent that its speed will decrease over halfan hour? (1) 1% (3) 48% () 79% (4) 150% 8. Over the last 10 years, the price of corn has decreased by 5% per bushel. (a) Assuming a steady percent decrease, by what percent does it decrease each year? Round to the nearest tenth of a percent. (b) Assuming this percent continues, by what percent will the price of corn decrease by after 50 years? Show the calculation that leads to your answer.. Round to the nearest percent.
9 9 LESSON #7  INTRODUCTION TO LOGARITHMS COMMON CORE ALGEBRA II Eponential functions are of such importance to mathematics that their inverses, functions that reverse their action, are important themselves. These functions, known as logarithms, will be introduced in this lesson. Eercise #1: The function y is shown graphed on the aes below along with its table of values. y y (a) Is this function onetoone? Eplain your answer. (b) Based on your answer from part (a), what must be true about the inverse of this function? (c) Create a table of values below for the inverse of y and plot this graph on the aes given. Notice that, as always, the graphs of a function and its inverse are symmetric across. y (d) What would be the first step to find an equation for this inverse algebraically? Write this step down and then stop. Defining Logarithmic Functions The function y log b is the name we give the inverse of y b. For eample, y log is the inverse of y. Based on Eercise #1(d), we can write an equivalent eponential equation for each logarithm as follows: y y log is the same as b b Based on this, we see that a logarithm gives as its output (yvalue) the eponent we must raise b to in order to produce its input (value).
10 Eercise #: Evaluate the following logarithms. If needed, write an equivalent eponential equation. Do as many as possible without the use of your calculator. (a) log 8 (b) log 416 (c) log5 65 (d) log10100, (e) log (f) log (g) log5 5 (h) log3 9 It is critically important to understand that logarithms give eponents as their outputs. We will be working for multiple lessons on logarithms and a basic understanding of their inputs and outputs is critical. Eercise #3: If the function y would represent its yintercept? (1) 1 (3) 8 () 13 (4) 9 log 8 9 was graphed in the coordinate plane, which of the following Eercise #4: Between which two consecutive integers must log3 40 lie? (1) 1 and (3) 3 and 4 () and 3 (4) 4 and 5 Calculator Use and Logarithms Most calculators only have two logarithms that they can evaluate directly. One of them, log10, is so common that it is actually called the common log and typically is written without the base 10. log log 10 (The Common Log) Eercise #5: Evaluate each of the following using your calculator. (a) log100 (b) log (c) log 10
11 11 INTRODUCTION TO LOGARITHMS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Which of the following is equivalent to y log7? (1) y 7 (3) 7 y () 7 y (4) y 1 7. If the graph of y 6 is reflected across the line y then the resulting curve has an equation of (1) y 6 (3) log6 y () y log6 (4) y 6 3. The value of log5 167 is closest to which of the following? Hint guess and check the answers. (1).67 (3) 4.58 () 1.98 (4) Which of the following represents the yintercept of the function y (1) 8 (3) 3 () 5 (4) 5 log ? 5. Determine the value for each of the following logarithms. (Easy) (a) log 3 (b) log7 49 (c) log36561 (d) log Determine the value for each of the following logarithms. (Medium) (a) log 1 64 (b) log3 1 (c) log5 1 5 (d) log
12 7. Determine the value for each of the following logarithms. Each of these will have noninteger, fractional answers. (Difficult) 3 5 (a) log4 (b) log4 8 (c) log 5 (d) log Between what two consecutive integers must the value of log lie? Justify your answer. 9. Between what two consecutive integers must the value of log lie? Justify your answer. APPLICATIONS 10. In chemistry, the ph of a solution is defined by the equation ph log H where H represents the concentration of hydrogen ions in the solution. Any solution with a ph less than 7 is considered acidic and any solution with a ph greater than 7 is considered basic. Fill in the table below. Round your ph s to the nearest tenth of a unit. Substance Concentration of Hydrogen ph Basic or Acidic? 7 Milk Coffee Bleach.5 10 Lemmon Juice Rain REASONING 11. Can the value of log 4 you about the domain of log b? be found? What about the value of log 0? Why or why not? What does this tell
13 13 LESSON #8  GRAPHS OF LOGARITHMS COMMON CORE ALGEBRA II The vast majority of logarithms that are used in the real world have bases greater than one; the ph scale that we saw on the last homework assignment is a good eample. In this lesson we will further eplore graphs of these logarithms, including their construction, transformations, and domains and ranges. Eercise #1: Consider the logarithmic function y log3 and its inverse y 3. (a) Construct a table of values for y 3 and then use this to construct a table of values for the function y log3. y y 3 y log 3 (b) Graph y 3 and y log3 on the grid given. Show the algebraic process of finding the inverse of y 3 to get y log3. (c) State the natural domain and range of y 3 and y log3. Domain: Range: y 3 Domain: Range: y log3 Eercise #: Using your calculator, sketch the graph of y log10 on the aes below. Label the intercept. State the domain and range of y log10. y Domain: Range: 10
14 14 Eercise #3: Given the function, y e, (a) Algebraically, find the function s inverse. Just as e is a very important number in eponential modeling, log e also arises in real world problems very often. It is called the natural logarithm and is usually epressed as ln. In other words, log ln e. Therefore, y e and y ln are inverses of each other. (b) Construct a table of values for y e and then use this to construct a table of values for the function y ln. Round values to the nearest tenth. y y e y ln Eercise #4: Without the use of your calculator, determine the values of each of the following. (a) ln e (b) 5 ln 1 (c) ln e (d) ln e Eercise #5: Which of the following equations describes the graph shown below? y y log 1 (1) 3 () y log 3 1 (3) y log 3 1 (4) y log 3 1 3
15 The fact that finding the logarithm of a nonpositive number (negative or zero) is not possible in the real number system allows us to find the domains of a variety of logarithmic functions. Eercise #6: Determine the domain of the function y log 3 4. State your answer in setbuilder notation. 15 All logarithms with bases larger than 1 are always increasing. This increasing nature can be seen by calculating their average rate of change. Eercise #7: Consider the common log, or log base 10, f log. y (a) Set up and evaluate an epression for the average rate of change of f over the interval 1 10 (b) Set up and evaluate an epression for the average rate of change of f over the interval (c) What do these two answers tell you about the changing slope of this function?
16 16 GRAPHS OF LOGARITHMS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. The domain of y log 5 in the real numbers is (1) 0 (3) 5 () 5 (4) 4 3. Which of the following equations describes the graph shown below? y (1) y log5 (3) y log3 () y log (4) y log 4 3. Which of the following represents the yintercept of the function y (1) 8 (3) 1 () 4 (4) 4 log 3 1? 4. Which of the following values of is not in the domain of f log 10 (1) 3 (3) 5 () 0 (4) 4 5. Which of the following is true about the function y (1) It has an intercept of 4 and a yintercept of 1. () It has intercept of 1 and a yintercept of 1. (3) It has an intercept of 16 and a yintercept of 1. (4) It has an intercept of 16 and a yintercept of 1. log 16 1? 4 5?
17 17 6. Which of the following is closest to the yintercept of the function whose equation is y 1 10e? (1) 10 (3) 7 () 18 (4) 5 7. On the grid below, the solid curve represents y e describe the dashed curve? Eplain your choice. (1) y 1 (3) y. Which of the following eponential functions could y () y e (4) y 4 8. Determine the domains of each of the following logarithmic functions. State your answers using any accepted notation. Be sure to show the inequality that you are solving to find the domain and the work you use to solve the inequality. (a) ylog 1 (b) ylog Graph the logarithmic function y log 4 on the graph paper given. For a method, see Eercise #1 on page 13. y
18 Without the use of your calculator, determine the values of each of the following. (a) ln e (b) 1 ln e 4 (c) ln e (d) 10 ln e e 10 REASONING 11. Logarithmic functions whose bases are larger than 1 tend to increase very slowly as increases. Let's investigate this for f log. (a) Find the value of f 1, f, f 4, and f 8 without your calculator. (b) For what value of will? For what value of will log 10 log 0?
19 19 LESSON #9  SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS COMMON CORE ALGEBRA II Eercise #1: Evaluate the following logarithms. (a) log 4 (b) log 4 3 (c) How many TIMES bigger was 3 log 4 compared to log 4? Therefore, log 4 3 log 4. This property can be generalized to logarithms of any base. 3 THE THIRD LOGARITHM LAW This law can be used with a logarithm of ANY base. The natural logarithm is used most often when solving eponential equations, which is what we will use in this course. Our general law using, better known as, is: In future courses, you will learn more about the reasons behind this law. For now, know that logarithms are equal to eponents. When a power (or eponent) is raised to another power, you multiply. Therefore when the argument of a logarithm is raised to a power, you can also multiply the logarithm by that power. Earlier in this unit, we used the Method of Common Bases to solve eponential equations. This technique is quite limited, however, because it requires the two sides of the equation to be epressed using the same base. A more general method utilizes our calculators and the third logarithm law: Eercise #: Solve: 4 8 using (a) common bases and (b) the logarithm law shown above. (a) Method of Common Bases (b) Logarithm Approach
20 The beauty of this logarithm law is that it removes the variable from the eponent. This law, in combination with the logarithm base e, the natural log, allows us to solve almost any eponential equation using calculator technology. Eercise #3: Solve each of the following equations for the value of. Round your answers to the nearest hundredth. (a) 5 18 (b) (c) These equations can become more complicated, but each and every time we will use the logarithm law to transform an eponential equation into one that is more familiar (linear only for now) Eercise #4: Solve each of the following equations for. Round your answers to the nearest hundredth. 3 (a) (b) Now that we are familiar with this method, we can revisit some of our eponential models from earlier in the unit. Recall that for an eponential function that is growing: If quantity Q is known to increase by a fied percentage p, in decimal form, then Q can be modeled by where represents the amount of Q present at and t represents time. Eercise #5: A biologist is modeling the population of bats on a tropical island. When he first starts observing them, there are 104 bats. The biologist believes that the bat population is growing at a rate of 3% per year. (a) Write an equation for the number of bats,, as a function of the number of years, t, since the biologist started observing them. (b) Using your equation from part (a), algebraically determine the number of years it will take for the bat population to reach 00. Round your answer to the nearest year.
21 Eercise #6: A stock has been declining in price at a steady pace of 5% per week. If the stock started at a price of $.50 per share, determine algebraically the number of weeks it will take for the price to reach $ Round your answer to the nearest week. 1 While we usually use the power rule and the natural log to solve eponential equations, you can also use a logarithm with the same base as the eponential equation. Eercise #7: Find the solution to each of the following eponential equations in terms of a logarithm with the same base as the eponential equation. Epress your answer first as a logarithm and then use the calculator to round to the nearest hundredth. (a) (b) Eercise #8: Find the solution to the general eponential equation ab c d, in terms of the constants a, c, d, and the logarithm of base b. Think about reversing the order of operations in order to solve for.
22 SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Which of the following values, to the nearest hundredth, solves: 7 500? (1) 3.19 (3).74 () 3.83 (4) The solution to 5, to the nearest tenth, is which of the following? (1) 7.3 (3) 11.4 () 9.1 (4) To the nearest hundredth, the value of that solves (1) 6.73 (3) 8.17 () 5.74 (4) is 4. Solve each of the following eponential equations. Round each of your answers to the nearest hundredth. (a) (b) (c) Solve each of the following eponential equations. Be careful with your use of parentheses. Epress each answer to the nearest hundredth. (a) (b).05t e (c)
23 6. Find the solution to each of the following eponential equations in terms of a logarithm with the same base as the eponential equation. Epress your answer first as a logarithm and then use the calculator to round to the nearest hundredth. 3 (a) (b) APPLICATIONS 6. The population of Red Hook is growing at a rate of 3.5% per year. If its current population is 1,500, in how many years will the population eceed 0,000? Round your answer to the nearest year. Only an algebraic solution is acceptable. 7. A radioactive substance is decaying such that % of its mass is lost every year. Originally there were 50 kilograms of the substance present. (a) Write an equation for the amount, A, of the substance left after tyears. (b) Find the amount of time that it takes for only half of the initial amount to remain. Round your answer to the nearest tenth of a year. REASONING 8. If a population doubles every 5 years, how many years will it take for the population to increase by 10 times its original amount? First: If the population gets multiplied by every 5 years, what does it get multiplied by each year? Use this to help you answer the question.
24 4 LESSON #30  COMPOUND INTEREST COMMON CORE ALGEBRA II In the worlds of investment and debt, interest is added onto a principal in what is known as compound interest. The percent rate is typically given on a yearly basis, but could be applied more than once a year. This is known as the compounding frequency. Let's take a look at a typical problem to understand how the compounding frequency changes how interest is applied. Eercise #1: A person invests $500 in an account that earns a nominal yearly interest rate of 4%. (a) How much would this investment be worth in 10 years if the compounding frequency was once per year? Show the calculation you use. (b) If, on the other hand, the interest was applied four times per year (known as quarterly compounding), why would it not make sense to multiply by 1.04 each quarter? (c) If you were told that an investment earned 4% per year, how much would you assume was earned per quarter? Why? (d) Using your answer from part (c), calculate how much the investment would be worth after 10 years of quarterly compounding? Show your calculation. So, the pattern is fairly straightforward. For a shorter compounding period, we get to apply the interest more often, but at a lower rate. Eercise #: How much would $1000 invested at a nominal % yearly rate, compounded monthly, be worth in 0 years? Show the calculations that lead to your answer. (1) $ (3) $ () $ (4) $ This pattern is formalized in a classic formula from economics that we will look at in the net eercise. Eercise #3: For an investment with the following parameters, write a formula for the amount the investment is worth, A, after tyears. P = amount initially invested r = nominal yearly rate n = number of compounds per year
25 Eercise #4: If $1500 is invested at.5% interest compounded weekly, how much will be in the account after 10 years to the nearest dollar. 5 Eercise #5: If $100 is invested at 8% interest compounded monthly, after how many years will the amount in the account double? Round to the nearest tenth of a year. The rate in was referred to as nominal (in name only). It's known as this, because you effectively earn more than this rate if the compounding period is more than once per year. Because of this, bankers refer to the effective rate, or the rate you would receive if compounded just once per year. Let's investigate this. Eercise #6: An investment with a nominal rate of 5% is compounded at different frequencies. Give the effective yearly rate, accurate to two decimal places, for each of the following compounding frequencies. Show your calculation. (a) Quarterly (b) Monthly (c) Daily
26 We could compound at smaller and smaller frequency intervals, eventually compounding all moments of time. In our formula from Eercise #6, we would be letting n approach infinity. Interestingly enough, this gives rise to continuous compounding and the use of the natural base e in the famous continuous compound interest formula. CONTINUOUS COMPOUND INTEREST For an initial principal, P, compounded continuously at a nominal yearly rate of r, the investment would be worth an amount A given by: 6 Eercise #7: A person invests $350 in a bank account that promises a rate of % continuously compounded. (a) Write an equation for the amount this investment would be worth after tyears. (b) How much would the investment be worth after 0 years? (c) Algebraically determine the time it will take for the investment to reach $400. Round to the nearest tenth of a year. (d) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Eercise #8: A population of 500 llamas on a tropical island is growing continuously at a rate of 3.5% per year. (a) Write a function to model the number of llamas on the island after tyears. (b) Algebraically determine the number of years for the population to reach 600. Round your answer to the nearest tenth of a year.
27 7 COMPOUND INTEREST COMMON CORE ALGEBRA II HOMEWORK APPLICATIONS 1. The value of an initial investment of $400 at 3% interest compounded quarterly can be modeled using which of the following equations, where t is the number of years since the investment was made? (1) A t (3) 4 A t () A t (4) A t. Which of the following represents the value of an investment with a principal of $1500 with a interest rate of.5% compounded monthly after 5 years? (1) $1, (3) $4,178. () $1, (4) $5, Franco invests $4,500 in an account that earns a 3.8% interest rate compounded continuously. If he withdraws the profit from the investment after 5 years, how much has he earned on his investment? (1) $858.9 (3) $9.50 () $91.59 (4) $ An investment that returns a 4.% yearly rate, but is compounded quarterly, has an effective yearly rate closest to (1) 4.1% (3) 4.7% () 4.4% (4) 4.3% 5. If an investment's value can be modeled with investment?.07 A t then which of the following describes the (1) The investment has a rate of 7% compounded every 1 years. () The investment has a rate of.7% compounded ever 1 years. (3) The investment has a rate of 7% compounded 1 times per year. (4) The investment has a rate of.7% compounded 1 times per year.
28 8 6. An investment of $500 is made at.8% nominal interest compounded quarterly. (a) Write an equation that models the amount A the investment is worth tyears after the principal has been invested. (b) How much is the investment worth after 10 years? (c) Algebraically determine the number of years it will take for the investment to be reach a worth of $800. Round to the nearest hundredth. (d) Why does it make more sense to round your answer in (c) to the nearest quarter? State the final answer rounded to the nearest quarter. 7. An investment of $300 is made at 3.6% nominal interest compounded continuously. (a) Write an equation that models the amount A the investment is worth tyears after the principal has been invested. (b) How much is the investment worth after 10 years? (c) Algebraically determine the number of years it will take for the investment to be reach a worth of $800. Round to the nearest hundredth. REASONING rt 8. The formula A Pe calculates the amount an investment earning a rate of r compounded continuously is worth. Show that the amount of time it takes for the investment to double in value is given by the epression ln. r
29 9 LESSON #31  MORE EXPONENTIAL AND LOGARITHMIC MODELING COMMON CORE ALGEBRA II The Halflife is the amount of time required for the amount of something to decrease to half its initial value. Any eponential decay function can be rewritten as a halflife function. We will begin with an eample from the first lesson of this unit: Eercise #1: The population of a town is decreasing by 4% per year and started with 1,500 residents. (a) Write a function to model this situation. (b) Algebraically determine when the population of the town will be half of the initial population? Round to the nearest tenth of a year. HALF LIFE FORMULA For an initial quantity, P, that is decreasing at an eponential rate, with a half life, h, the amount of the quantity, A, left after t time units is given by the formula, (c) Write the halflife formula for the population of the town for Eercise #1. Eercise #: The decay of a sample of 5000 grams of carbon can be modeled by the equation, t Ct ( ) 5000, where t is measured in years. (a) What is the halflife of carbon? (b) How can you tell this is a halflife equation?
30 30 Eercise #3: Copper has antibacterial properties, and it has been shown that direct contact with copper alloy C11000 at 0 C kills methicillinresistant Staphylococcus aureus (MRSA) over a period of time. (a) A function that models a population of 1,000 MRSA bacteria t minutes after coming in contact with copper alloy C11000 is A(t)=1000(.91) t. What does the base 0.91 mean in this scenario? (b) Rewrite the formula for A as an eponential function with base of 1 (Halflife Formula). Round the halflife to the nearest hundredth of a minute. (c) Convert your equation back into its original form. There are a number of other eponential models, but these equations will be given. Below is an eample. Eercise #4: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled using the eponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of the number of minutes, m, it has been cooling. (a) What was the initial temperature of the water at m 0. Do without using your calculator. T ( m) = 101e 0.03m + 67 (b) How do you interpret the statement that T ? (c) Determine algebraically when the temperature of the liquid will reach 100 F. Show the steps in your solution. Round to the nearest tenth of a minute. (d) On average, how many degrees are lost per minute over the interval 10 m 30? Round to the nearest tenth of a degree.
31 Logarithmic functions can also be used to model real world phenomena. Eample #5: The slope, s, of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach by this equation: s log d. s (a) Sand particles typically have a maimum diameter of 1mm. Using this information, sketch a graph of the function. 31 d (b) If the average diameter of the sand particles is 0.5mm, find the slope of the beach (to the nearest hundredth). (c) Given a slope of 0.14, find the average diameter (to the nearest hundredth) of the sand particles on the beach. Eample #6: Two methods of instruction were used to teach athletes how to shoot a basketball. The methods were assessed by assigning students into two groups, one who was taught with method A and one who was taught with method B. The students in each group took 30 foul shots after each of ten sessions. The average number of shots made in each of the sessions by an athlete using method A can be modeled by the function, A( ) ln. The average number of foul shots made in each of the sessions for an athlete using method B can be modeled by the function, B ( ) 9.17(1.109). (a) In which of the 10 sessions, to the nearest whole number, will the two methods produce the same number of made baskets? Eplain how you found your answer. (b) Find the average range of change for each method between sessions 3 and 8 for each of the methods. Give proper units and round your answers to the nearest tenth. (c) Eplain why B() would not be an appropriate model for this situation if there were 15 sessions.
32 MORE EXPONENTIAL AND LOGARITHMIC MODELING COMMON CORE ALGEBRA II HOMEWORK 1. The decay of a sample of 800 grams of hydrogen can be modeled by the equation, t years. (a) What is the halflife of hydrogen? t 1 Ht ( ) after (b) How can you tell this is a halflife equation?. The $,500 in your bank account is decreasing continuously at a rate of 5% per year. (a) Use the Continuous Compound Interest Formula to write a function that models the amount of money in your bank account after t years. (Don t forget the r value should be negative because it s decreasing). (b) When will only half of your initial deposit be left in your bank account (to the nearest tenth of a year)? (c) Write the halflife formula for your bank account. 3. Flu is spreading eponentially at a school. The number of new flu patients can be modeled using the equation 0.1d F 10e, where d represents the number of days since 10 students had the flu. (a) How many days will it take for the number of new flu patients to equal 50? Round your answer to the nearest day. (b) Find the average rate of change of F over the first three weeks, i.e. 0 d 1. Show the calculation that leads to your answer. Give proper units and round your answer to the nearest tenth. What is the physical interpretation of your answer?
33 4. Jessica keeps track of the height of a tree she planted over the first ten years. It can be modeled by the equation y ln( 1) where is the number of years since she planted the tree. (a) On average, how many feet did the tree grow each year over the time interval 0 t 10, to the nearest hundredth. 33 (b) How tall was the tree when she planted it? 5. Most tornadoes last less than an hour and travel less than 0 miles. The wind speed s (in miles per hour) near the center of a tornado is related to the distance d (in miles) the tornado travels by this model: s = 93logd +65. (a) Sketch a graph of this function. y (b) On March 18, 195, a tornado whose wind speed was about 180 miles per hour struck the Midwest. Use your graph to determine how far the tornado traveled to the nearest mile.
34 34 LESSON #3  NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW COMMON CORE ALGEBRA II NEWTON S LAW OF COOLING where: T(t) is the temperature of the object t time units has elapsed, T a is the ambient temperature (the temperature of the surroundings), assumed to be constant, not impacted by the cooling process, T 0 is the initial temperature of the object, and k is the decay constant per time unit (the r value where r is negative). Eercise #1: A detective is called to the scene of a crime where a dead body has just been found. He arrives at the scene and measures the temperature of the dead body at 9:30 p.m to be 7 F. After investigating the scene, he declares that the person died 10 hours prior, at approimately 11:30 a.m. A crime scene investigator arrives a little later and declares that the detective is wrong. She says that the person died at approimately 6:00 a.m., hours prior to the measurement of the body temperature. She claims she can prove it by using Newton s law of cooling. Using the data collected at the scene, decide who is correct, the detective or the crime scene investigator. T a = 68 F (the temperature of the room) T 0 = 98.6 F (the initial temperature of the body) k = (13.35 % per hour calculated by the investigator from the data collected) Recall, the temperature of the body at 9:30 p.m. is 7 F. Eercise : A detective is called to the scene of a crime where a dead body has just been found. She arrives on the scene at 10:3 pm and begins her investigation. Immediately, the temperature of the body is taken and is found to be 80 o F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68 o F. Assuming that the victim s body temperature was normal (98.6 o F) prior to death and that the temperature of the victim s body decreases continuously at a rate of 13.35% per hour, use Newton s Law of Cooling to determine the time when the victim died.
35 Eercise #3: Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 180 F and k is (for time in minutes). Suppose both cups are poured at the same time. Cup 1 is left sitting in the room that is 75 F, and cup is taken outside where it is 4 F. i. Use Newton s law of cooling to write equations for the temperature of each cup of coffee after t minutes have elapsed. 35 ii. Graph and label both on the same coordinate plane and compare and contrast the end behavior of the two graphs. iii. Coffee is safe to drink when its temperature is below 140 F. How much time elapses before each cup is safe to drink, to the nearest tenth of a minute. Use a graph to answer the question.
36 36 Throughout the unit, you have learned many different eponential formulas. We will now practice writing a few of them and converting between the different forms. Eercise #4: Tritium has a halflife of 1.3 years. a. Write a halflife formula, A(t), for the amount of tritium left in a 500 milligram sample after t years. b. Write an equivalent function, B(t), it terms of the yearly rate of decay of tritium. Round all values to four decimal places. c. Wrtie an equivalent function, C(t), it terms of the monthly rate of decay of tritium. Round all values to four decimal places. Eercise #5: A deposit of $300 is made into a bank account that gets 4.3% interest compounded continuously. a. Write a function, A(t), to model the amount of money in the account after t years. b. Write an equivalent function, B(t), it terms of the yearly rate of interest for the account. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the quarterly rate of interest for the account. Round all values to four decimal places.
37 37 NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW COMMON CORE ALGEBRA II HOMEWORK 1. Hot soup is poured from a pot and allowed to cool in a room. The temperature in degrees Fahrenheit of the soup after t minutes, can be modeled by the function, T(t)= 65+(165)e .054t. What was the initial temperature of the soup? What is the temperature of the room? At what rate is the temperature of the soup decreasing?. Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 190 F and k is (for time in minutes). Both are left sitting in the room that is 75 F, but milk is immediately poured into cup cooling it to an initial temperature of 16 F. a. Use Newton s law of cooling to write equations for the temperature of each cup of coffee after t minutes have elapsed. b. Graph and label both functions on the coordinate plane and compare and contrast the end behavior of the two graphs. c. Coffee is safe to drink when its temperature is below 140 F. Based on your graph, how much time elapses before each cup is safe to drink to the nearest tenth of a minute?
38 3. A cooling liquid starts at a temperature of 00 F and cools down in a room that is held at a constant temperature of 70 F. (Note time is measured in minutes on this problem). 38 (a) Use Newton s Law of Cooling to determine the value of k if the temperature after 5 minutes is to four decimal places. (Hint: Write out the equation, plug in (5,153), and solve for k.). Round (b) Using the value of k you found in part (a), algebraically determine, to the nearest tenth of a minute, when the temperature reaches 100 F. 4. A deposit of $100 is made into a bank account that gets 4.3% interest compounded weekly. a. Write a function, A(t), to model the amount of money in the account after t years. b. Write an equivalent function, B(t), it terms of the yearly rate of interest for the account. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the monthly rate of interest for the account. Round all values to four decimal places.
39 39 5. A small town has a population of 1,600. The population is decreasing continuously at a rate of 3% per year. a. Write a function, A(t), to model the population of the the town after t years. b. Write an equivalent function, B(t), it terms of the yearly rate of decrease for the town. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the daily rate of decrease for the town. Round all values to four decimal places.
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