LESSON #56 - WRITING EQUATIONS OF EXPONENTIAL FUNCTIONS COMMON CORE ALGEBRA II

1 LESSON #56 - WRITING EQUATIONS OF EXPONENTIAL FUNCTIONS COMMON CORE ALGEBRA II One of the skills that you acquired in Common Core Algebra I was the ability to write equations of exponential functions if you had information about the starting value and base (multiplier or growth constant). Let's review a very basic problem. Exercise #1: An exponential function of the form f x a b x is presented in the table below. Determine the values of a and b and explain your reasoning. a b x 0 1 2 3 f x 5 15 45 135 Final Equation: Explanation: Finding an exponential equation becomes much more challenging if we do not have output values for inputs that are increasing by unit values (increasing by 1 unit at a time). Let's start with a basic problem. Exercise #2: For an exponential function of the form f x a b x, it is known that f 0 8 and f. f 0 8 to determine the value of a. Show your thinking. (a) Use the fact that 3 1000 (b) Use you answer from part (a) and the fact that f 3 1000 to set up an equation to solve for b. You will solve for b in part (c). (c) Solve for the value of b using properties of exponents. (d) What is the equation of the exponential function, f(x)?

Now, let's work with the most generic type of problem. Just like with lines, any two (non-vertically aligned) points will uniquely determine the equation of an exponential function. Exercise #3: An exponential function of the form (a) By substituting these two points into the general form of the exponential, create a system of equations in the constants a and b. y a b x passes through the points 2, 36 and 5,121.5. (b) Divide these two equations to eliminate the constant a. Recall that when dividing to like bases, you subtract their exponents. 2 (c) Solve the resulting equation from (b) for the base, b. (d) Use your value from (c) to determine the value of a. State the final equation. Let's now get some practice on this with a decreasing exponential function. Exercise #4: Find the equation of the exponential function shown graphed below. Be careful in terms of your x exponent manipulation. State your final answer in the form y a b. y Exercise #5: A bacterial colony is growing at an exponential rate. It is known that after 4 hours, its population is at 98 bacteria and after 9 hours it is 189 bacteria. Determine an equation in x y a b form that models the population, y, as a function of the number of hours, x. Round all values to the nearest hundredth. At what percent rate is the population growing per hour, to the nearest percent? x

3 LESSON #56 - FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Find the equation of the exponential function, in the form x points. Show the work that you use to arrive at your answer. 0,10 and 3, 80 y a b that passes through the two coordinate 2. For each of the following coordinate pairs, find the equation of the exponential function, in the form y a b x that passes through the pair. Show the work that you use to arrive at your answer. (a) 2,192 and 5,12288 (b) 1,192 and 5, 60.75 3. Each of the previous problems had values of a and b that were rational numbers. They do not need not be. Find the equation for an exponential function that passes through the points 2,14 and 7, 205 in y a b x form. When you find the value of b do not round your answer before you find a. Then, find both to the nearest hundredth and give the final equation. Check to see if the points fall on the curve.

Water Depth (ft) 4 APPLICATIONS 4. A population of koi goldfish in a pond was measured over time. In the year 2002, the population was recorded as 380 and in 2006 it was 517. Given that y is the population of fish and x is the number of years since 2000, do the following: (a) Represent the information in this problem as two coordinate points. (b) Determine an exponential function of the form x y a b that passes through these two points. Round b to the nearest hundredth and a to the nearest tenth. (c) Use your function to predict the population of fish in the year 2000. Justify your work. 5. Engineers are draining a water reservoir until its depth is only 10 feet. The depth decreases exponentially as shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours to be 28 feet. The engineers found the exponential function y 84.31(0.76) x to model the depth of the water after x hours. Graph the horizontal line y 10 and find its intersection to determine the time, to the nearest tenth of an hour, when the reservoir will reach a depth of 10 feet. Time (hrs)

5 LESSON #57 - EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II Exponential functions are very important in modeling a variety of real world phenomena because certain things either increase or decrease by fixed 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. Exercise #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 first 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 2 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 Exercise #1 can be generalized to any situation where a quantity is increased by a fixed percentage over a fixed interval of time. This pattern is summarized below: INCREASING EXPONENTIAL MODELS If quantity A is known to increase by a fixed percent rate, r, in decimal form, then A can be modeled by where represents the amount of A present at and t represents time. Exercise #2: Which of the following gives the savings S in an account if $250 was invested at an interest rate of 3% per year? (1) S 250 4 t t (3) S 1.03 250 (2) S 250 1.03 t (4) S 250 1.3 t

Decreasing exponentials are developed in the same way, but have a negative percent which is still 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. Exercise #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) 2% (c) 25% (d) 0.5% 6 DECREASING EXPONENTIAL MODELS If Quantity A is known to decrease by a fixed percent rate, then r, in decimal form, has a negative value. A can still be modeled by where represents the amount of A present at and t represents time. Exercise #4: If the population of a town is decreasing by 4% per year and started with 12,500 residents, which of the following is its projected population in 10 years? Show the exponential model you use to solve this problem. (1) 9,230 (3) 18,503 (2) 76 (4) 8,310 Exercise #5: For each equation, a) Identify the problem as growth or decay b) Name P, the starting amount (Principle) c) Find r, the percent change and indicate if it is an increase or decrease. 1) gt ( ) 900(.88) t 2) f( t) 7(1.345) t 3) ht ( ) 420(2.3) t 4) qx ( ) 5000(.9925) x

Exercise #5: The stock price of WindpowerInc is increasing at a rate of 4% per week. Its initial value was $20 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 $120. (a) Model both stock prices using exponential functions. 7 (b) Then, find when the stock prices will be equal graphically, to the nearest week. Draw a well labeled graph to justify your solution.

8 APPLICATIONS LESSON #57 BASIC EXPONENTIAL 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) $218 (3) $168 (2) $192 (4) $324 2. 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) 12 (2) 6 (4) 28 3. If a radioactive substance is quickly decaying at a rate of 13% per hour approximately how much of a 200 pound sample remains after one day? (1) 7.1 pounds (3) 25.6 pounds (2) 2.3 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) 257 (3) 102 (2) 58 (4) 189 5. Which of the following equations would model a population with an initial size of 625 that is growing at an annual rate of 8.5%? t P (3) P 1.085 625 (1) 625 8.5 t (2) P 625 1.085 t (4) P 2 8.5t 625 6. 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, which of the following equations best models the acceleration t seconds after the object begins falling? (1) (2) a a 2 15 9.8t (3) a 9.8 1.15 t 9.8 15t (4) a 9.8 0.85 t

9 7. For each equation, a) Identify the problem as growth or decay b) Name P, the starting amount (Principle) c) Find r, the percent change and indicate if it is an increase or decrease. 1) gt ( ) 325(1.13) t 2) f( t) 75(.965) t 3) ht ( ) 40(.79) t 4) qx ( ) 420(3.7) x 8. Red Hook has a population of 6,200 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. After how many years, will Red Hook have a greater population than Rhinebeck? Write an exponential function for the growth of each town, and solve the problem using a table. 9. A warm glass of water, initially at 120 degrees Fahrenheit, is placed in a refrigerator at 34 degrees Fahrenheit and its temperature is seen to decrease according to the exponential function (a) Verify that the temperature starts at 120 degrees Fahrenheit by evaluating T 0. h T h 86 0.83 34 (b) Using your calculator, sketch a graph of T below for all values of h on the interval 0 h 24. Be sure to label your y-axis and y- intercept. (c) Use the graph to determine 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).

10 REASONING 10. 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? difference Hint: To calculate percent growth use the formula 100 original amount.

11 LESSON #58 - SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMIC FORM COMMON CORE ALGEBRA II Earlier in this unit, we used the Method of Common Bases to solve exponential equations. This technique is quite limited, however, because it requires the two sides of the equation to be expressed using the same base. A more general method utilizes logarithmic form and the calculator to solve these problems. Exercise #1: Solve: 4 x 8 using (a) common bases and (b) logarithmic form. (a) Method of Common Bases (b) Logarithmic Form The next example cannot be solved with common bases. Exercise #2: Find x to the nearest hundredth: 2 x 11 Steps: 1. Use SADMEP to isolate the exponential part if necessary. 2. Write the equation to logarithmic form. 3. Use your calculator to evaluate the logarithm. Note: If you get a repeating decimal at some point while you are solving the problem, be sure to copy and paste it to keep the entire answer. Otherwise your final answer might be slightly off. Exercise #3: Solve each of the following equations to the nearest hundredth. (a) 116. x 12 7. 7 (b) 4 5 47.6 x

12 (c) 20,000 1000 1.08 12t (d) 5(1.06) x 150 (e) x 4 2 3 17 (f) 3 2 5 x 102 Exercise #4: The money in your bank account, which is currently $527, decreases by 4% each month. a) Write an exponential function that models this situation. b) How much money will you have in five months? Round to the nearest dollar. c) Algebraically determine how long will it take until you only have $100 left in the account? Round to the nearest month. Exercise #5: Find the solution to the general exponential equation a b cx 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 x.

13 LESSON #58 - SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMIC FORM COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Solve each of the following equations. Round your answers to the nearest hundredth. (a) 10 x 50 (b) 4(3) x 72 b (c) 75 4 20 (d) 2500 2000 1.05 t (e) 5 10 6 100 x (f) 4 x 13 2 117 (e) 6 3 4 x 2 100 (f) 400(3) 10,500 x

14 REASONING x 2. Given the equation, e 10, answer the questions that follow. (a) Solve the equation. Round your answer to the nearest tenth. (b) What is the name of the logarithm used in part (a)? APPLICATION 3. A small country whose current population in 2010 is 300,000 people, has been experiencing a 10% population increase every year. a) Write an exponential function that models this situation. b) To the nearest person, what will the population be in 2018? c) If the trend continues, in what year will the population reach 1 million people?

15 LESSON #59 PERIODIC EXPONENTIAL GROWTH AND DECAY 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. Exercise #1: A person invests $500 in an account that earns a yearly interest rate of 4%. (a) How much, to the nearest cent, 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. Exercise #2: How much would $1000 invested at a 2% yearly rate, compounded monthly, be worth in 20 years? Show the calculations that lead to your answer. (1) $1485.95 (3) $1033.87 (2) $1491.33 (4) $1045.32 This pattern is formalized in a classic formula from economics that we will look at in the next exercise. Exercise #3: For an investment with the following parameters, write a formula for the amount the investment is worth, A, after t-years. P = amount initially invested r = yearly rate n = number of compounds per year

Exercise #4: If $1500 was invested in 2010 at 2.5% interest per year, compounded weekly, how much will be in the account in 2020, to the nearest dollar. 16 Exercise #5: If $100 is invested at 8% yearly interest compounded monthly, after how many years will the amount in the account double? Round to the nearest tenth of a year. Exercise #6: A bank account starts with $3000. The amount in the account decreases at a rate of 5.5% per year, compounded daily. How much will be in the account after 4 years? Exercise #7: Matt bought a new car for $25,000 in 2016. The car depreciates approximately 15% of its value per year, compounded quarterly. In what year will the car be worth half of its initial value?

17 APPLICATIONS LESSON #59 PERIODIC EXPONENTIAL GROWTH AND DECAY COMMON CORE ALGEBRA II HOMEWORK 1. The value of an initial investment of $400 at 3% interest per year, 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 400 1.0075 4t (3) 4 A 400 1.03 t (2) A 400 1.0075 t (4) A 4 400 1.0303 t 2. Which of the following represents the value of an investment with a principal of $1500 with a yearly interest rate of 2.5% compounded monthly after 5 years? (1) $1,697.11 (3) $4,178.22 (2) $1,699.50 (4) $5,168.71 3. If an investment's value can be modeled with investment?.027 A 325 1 12 12t then which of the following describes the (1) The investment has a yearly rate of 27% compounded every 12 years. (2) The investment has a yearly rate of 2.7% compounded ever 12 years. (3) The investment has a yearly rate of 27% compounded 12 times per year. (4) The investment has a yearly rate of 2.7% compounded 12 times per year. 4. In 2000, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 33% per year, compounded weekly. In what year did the number of cell phone subscribers reach 30,000?

18 5. An investment of $500 is made at 2.8% yearly interest compounded quarterly. (a) Write an equation that models the amount A the investment is worth t-years 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. 6. There were 18,000 people who owned a basic DVD player in Kingston in the year 2010. This number is decreasing at a yearly rate of 6%, compounded monthly. In what year will the number of people who own a basic DVD player in Kingston decrease to 10,000? 7. James invests $1 in a bank account. This is an extremely profitable account that grows 100% interest per year! How much money would be in the account after 1 year with the following compounding rates: (a) Compounded once per year (b) Monthly (c) Daily (d) Hourly (there are 8760 hours in a year) (e) Your answers are approaching a famous number. Do you know what it is?

19 LESSON #60 CONTINUOUS EXPONENTIAL GROWTH AND DECAY & HALF LIFE COMMON CORE ALGEBRA II From the last lesson, we could compound at smaller and smaller frequency intervals, eventually compounding all moments of time. For example, we could let n approach infinity in the last homework problem. 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 amount, P, compounded CONTINUOUSLY at a yearly rate of r, the investment would be worth an amount A given by: Exercise #1: A person invests $350 in a bank account that promises a yearly rate of 2% continuously compounded. (a) Write an equation for the amount this investment would be worth after t-years. (c) Algebraically determine the time it will take for the investment to reach $400. Round to the nearest tenth of a year. (b) How much would the investment be worth after 20 years? Exercise #2: 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 t-years. (b) Algebraically determine the number of years for the population to reach 600. Round your answer to the nearest tenth of a year.

20 Exercise #3: A chemical compound with a weight of 300 grams decays at a rate of 1% per year, compounded continuously. (a) Write an equation to model the decay of the substance after t years. (c) Algebraically determine the time it will take for the half of the initial amount to remain. Round to the nearest tenth of a year. (b) How much of the substance remains after 10 years, to the nearest gram. The Half-life is the amount of time required for the amount of something to decrease to half its initial value. Any exponential decay function can be rewritten as a half-life function. HALF LIFE FORMULA For an initial quantity, P, that is decreasing at an exponential rate, with a half life, h, the amount of the quantity, A, left after t time units is given by the formula, Exercise #4: We will begin with the chemical compound in exercise #3. (a) What was the half-life of the substance? (b) Write a half life function to model the decay of the chemical compound. (c) Graph your answer to Exercise #3a and Exercise #4b using Xmin:0, Xmax: 200, Ymin:0, Ymax:300. What do you notice?

Exercise #5: The population of a town with 12,500 residents is decreasing at an exponential rate with a half-life of 17 years. (a) Write a function to model this situation. 21 (b) Algebraically determine when the population of the town will reach 5,000 people to the nearest tenth of a year. Exercise #6: The decay of a sample of 5000 grams of carbon can be modeled by the equation, t 1 5730 Ct ( ) 5000, where t is measured in years. 2 (a) What is the half-life of carbon? (b) How can you tell this is a half-life equation?

22 LESSON #60 CONTINUOUS EXPONENTIAL GROWTH AND DECAY & HALF LIFE COMMON CORE ALGEBRA II HOMEWORK 1. Franco invests $4,500 in an account that earns a 3.8% yearly interest rate compounded continuously. If he withdraws the profit from the investment after 5 years, how much has he earned on his investment? (1) $858.92 (3) $922.50 (2) $912.59 (4) $941.62 2. An investment of $300 is made at 3.6% yearly interest compounded continuously. (a) Write an equation that models the amount A the investment is worth t-years 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. 3. Dishwashers depreciate at a rate of 12.5% per year, compounded continuously. A $675 dishwasher was bought in 2016. (a) Write an function that models the value of the dishwasher t years after 2016. (b) How much is the dishwasher worth in 2020? (c) Algebraically determine the year in which the dishwasher will cost $200 to the nearest year.

4. The decay of a sample of 800 grams of hydrogen can be modeled by the equation, t years. t 1 Ht ( ) 800 2 12.32 23 after (a) What is the half-life of hydrogen? (b) How can you tell this is a half-life equation? 5. Copper has antibacterial properties, and it has been shown that direct contact with a copper alloy kills MRSA over a period of time. The MRSA is killed exponentially with a half-life of 7.52 minutes. (a) A 1000 MRSA bacteria come in contact with copper. Write a function to model the decay of the MRSA after t minutes. (b) How many MRSA bacteria remain after one hour, to the nearest whole number? 6. The $2,500 in your bank account is decreasing continuously at a rate of 5% per year. (a) Write a function that models the amount of money in your bank account after t years. (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 half-life formula for your bank account.

24 REASONING rt 7. 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 expression ln 2. r

25 LESSON #61 CONVERTING BETWEEN EXPONENTIAL GROWTH AND DECAY FORMULAS COMMON CORE ALGEBRA II Throughout the unit, you have learned many different exponential formulas. We will now practice writing a few of them and converting between the different forms. It is always easiest to find the equivalent basic growth and decay formula first and work from that formula. Exercise #1: A deposit of $800 is made into a bank account that gets 5.2% basic yearly interest. 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 monthly rate of interest for the account. Round all values to four decimal places. Identify the monthly rate of growth. c. Write an equivalent function, C(t), it terms of the daily rate of interest for the account. Round all values to four decimal places. Identify the daily rate of growth. Exercise #2: A microwave depreciates at a rate of 10% per year, compounded quarterly. a. Write a function, A(t), for the value of a microwave t years after it was bought. (Use the variable, P, for the starting value since it is unknown). b. Write an equivalent function, B(t), for the basic yearly rate of depreciation of the microwave. Round all values to four decimal places. What is the basic yearly rate of depreciation? c. Write an equivalent function, C(t), for the monthly rate of depreciation of the microwave. Round all values to four decimal places. What is the basic monthly rate of depreciation?

26 Exercise #3: 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), for the basic yearly rate of growth for the account. Round all values to four decimal places. What is the basic yearly rate of growth? c. Write an equivalent function, C(t), for the quarterly rate of growth for the account. Round all values to four decimal places. What is the quarterly rate of growth? Exercise #4: Tritium has a half-life of 12.32 years. a. Write a half-life formula, A(t), for the amount of tritium left in a 500 milligram sample after t years. b. Write an equivalent function, B(t), for the basic yearly rate of decay of tritium. Round all values to four decimal places. What is the basic yearly rate of decay? c. Write an equivalent function, C(t), for the monthly rate of decay of tritium. Round all values to four decimal places. What is the monthly rate of decay?

27 LESSON #61 CONVERTING BETWEEN EXPONENTIAL GROWTH AND DECAY FORMULAS COMMON CORE ALGEBRA II HOMEWORK APPLICATIONS 1. A deposit of $1200 is made into a bank account that gets 3.7% 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), for the basic yearly rate of interest for the account. Round all values to four decimal places. What is the basic yearly rate of growth? c. Write an equivalent function, C(t), for the monthly rate of interest for the account. Round all values to four decimal places. What is the basic monthly rate of growth? 2. A small town has a population of 12,600. The population is decreasing continuously at a rate of 12% 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), for the basic yearly rate of decrease for the town. Round all values to four decimal places. At what basic yearly rate is the population decreasing? c. Write an equivalent function, C(t), for the daily rate of decrease for the town. Round all values to four decimal places. At what basic daily rate is the population decreasing?

3. Cobalt-60 has a half-life of 5.27 years. a. Write a half-life formula, A(t), for the amount of cobalt-60 that remains after t years. (Use the variable, P, for the starting value since it is unknown). 28 b. Write an equivalent function, B(t), for the basic yearly rate of decay of cobalt-60. Round all values to four decimal places. What is the basic yearly rate of decay? c. Write an equivalent function, C(t), for the bi-annual rate of decay of cobalt-60. Round all values to four decimal places. What is the bi-annual rate of decay?

29 LESSON #62 - 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. Exercise #1: A population of wombats is growing at a constant percent rate. The population on January 1 st is 1027 and a year later is 1079. a. Assuming exponential growth, what is its annual growth rate to the nearest tenth of a percent? b. Assuming exponential growth, what is the monthly growth rate to the nearest tenth of a percent? c. Assuming exponential growth, what is the five-year growth rate to the nearest tenth of a percent? Exercise #2: A house purchased 10 years ago for $120,000 was just sold for $195,000. a. Assuming exponential growth, what is its ten-year growth rate to the nearest tenth of a percent? b. Assuming exponential growth, what is the two-year growth rate to the nearest tenth of a percent? c. Assuming exponential growth, what is the 20-year growth rate to the nearest percent? Exercise #3: A car purchased 3 years ago for $45,000 was just sold for $18,000. a. Assuming exponential growth, what is its three year rate of decay to the nearest tenth of a percent? b. Assuming exponential growth, what is the one year rate of decay to the nearest tenth of a percent? c. Assuming exponential growth, what is the weekly rate of decay to the nearest tenth of a percent?

Exercise #4: If a population was growing at a constant rate of 22% every 5 years, then what is its percent growth rate over at 2 year time span? Round to the nearest tenth of a percent. (a) Find the one-year percent growth rate first. (b) Find the two-year percent growth rate. 30 (c) How could you go from the five-year growth rate to the two-year growth rate in one step? Exercise #5: World oil reserves (the amount of oil unused in the ground) are depleting at a constant 2% per year. Determine what the percent decline will be over the next 20 years based on this 2% yearly decline. Exercise #6: A radioactive substance s half-life 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 half-life of 20 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.

31 LESSON #62 - 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% (2) 52% (4) 63% 2. 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) 0.135 12 (2) 1.135 12 (3) 1.135 12 (4) 1 12 1.135 3. The population of Saugerties is growing exponentially. If there were 21,000 people in Saugerties in 2015 and 21,420 people in Saugerties in 2016, answer the following questions. a. What is the yearly percent growth rate to the nearest tenth of a percent? b. What is the monthly growth rate to the nearest tenth of a percent? c. What is the 10-year growth rate to the nearest percent? 4. A fridge purchased 6 years ago for $1100 is now worth $730. a. Assuming exponential growth, what is its six-year rate of decay to the nearest tenth of a percent? b. Assuming exponential growth, what is the one-year rate of decay to the nearest tenth of a percent?

32 5. The county debt is growing at an annual rate of 3.5%. What percent rate is it growing at per 2 years? Per 5 years? Per decade? Show the calculations that lead to each answer. Round each to the nearest tenth of a percent. 6. A population of llamas is growing at a constant yearly rate of 6%. At what rate is the llama population growing per month? Round to the nearest tenth of a percent. 7. 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. 8. 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 expression that would calculate its growth rate after a single year. 9. 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 half-an hour? (1) 21% (3) 48% (2) 79% (4) 150%

33 LESSON #63 - MORE EXPONENTIAL AND LOGARITHMIC MODELING COMMON CORE ALGEBRA II We have been working with a number of exponential models with specific equations. There are a number of other exponential models, but these equations will be given. Below is an example. Exercise #1: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled using the exponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of the number of minutes, m, it has been cooling. T m ( ) = 101e -0.03m + 67 (a) What was the initial temperature of the water at m 0. Do without using your calculator. (b) How do you interpret the statement that T 60 83.7? (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. Logarithmic functions can also be used to model real world phenomena. Exercise #2: 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 0.159 0.118log d. (a) Sand particles typically have a maximum diameter of 1mm. Using this information, graph the function. (b) If the average diameter of the sand particles is 0.25mm, 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.

Exercise #3: Two methods of instruction were used to teach athletes how to shoot a basketball. The methods were assessed by randomly assigning students into two groups, one that was taught with method A and one that 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 x sessions by an athlete using method A can be modeled by the function, A( x) 11.90 4.3ln x. The average number of foul shots made in each of the x sessions for an athlete using method B can be modeled by the function, Bx ( ) 9.17(1.109) x. (a) Sketch a graph of both functions on the grid where 1 x 10. 34 (b) In which of the 10 sessions, to the nearest whole number, will the two methods produce the same number of made baskets? Explain how you found your answer. (c) 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. (d) Explain why B(x) would not be an appropriate model for this situation if there were 15 sessions.

35 LESSON #63 - MORE EXPONENTIAL AND LOGARITHMIC MODELING COMMON CORE ALGEBRA II HOMEWORK 1. The flu is spreading exponentially at a school. The number of flu patients can be modeled using the equation 0.12d F 10e, where d represents the number of days since 10 students had the flu. (a) Graph the function on the grid to the right over the first two weeks, 0 d 14. (b) How many days will it take for the number of new flu patients to equal 40? Round your answer to the nearest day. (c) Find the average rate of change of F over the first three weeks, i.e. 0 d 21. 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? 2. Jessica keeps track of the height of a tree, in feet, she planted over the first ten years. It can be modeled by the equation y 4.25 10.16ln( x 1) where x 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. (b) How tall was the tree when she planted it?

3. Most tornadoes last less than an hour and travel less than 20 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 36 (a) Sketch a graph of this function. s d (b) On March 18, 1925, 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.

37 LESSON #64 - NEWTON'S LAW OF COOLING 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). Exercise #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 72 F. After investigating the scene, he declares that the person died 10 hours prior, at approximately 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 approximately 6:00 a.m., 15. 5 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 = 0.1335 (13.35 % per hour calculated by the investigator from the data collected) Recall, the temperature of the body at 9:30 p.m. is 72 F. Exercise 2: A detective is called to the scene of a crime where a dead body has just been found. She arrives on the scene at 10:23 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.

Exercise #3: Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 180 F and k is 0.2337 (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 2 is taken outside where it is 42 F. i. Use Newton s law of cooling to write equations for the temperature of each cup of coffee after t minutes have elapsed. 38 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.

39 LESSON #64 - NEWTON'S LAW OF COOLING 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+(212-65)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? 2. Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 190 F and k is 0.1450 (for time in minutes). Both are left sitting in the room that is 75 F, but milk is immediately poured into cup 2 cooling it to an initial temperature of 162 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?

3. A cooling liquid starts at a temperature of 200 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). 40 (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.