Algebra I Block Unit #2: Sequences & Exponential Functions Lesson #5: The Power of Exponential Growth Name Period Date DAY #1 Ex #1: Two equipment rental companies have different penalty policies for returning a piece of equipment late. Company 1 charges $5 every day the equipment is returned late. Company 2 charges 1 cent for the first day, and doubles the penalty for every day the equipment is late. Jim rented a digger from Company 2 because he thought it had the better late return policy. The job he was doing with the digger took longer than he expected, but it did not concern him because the late penalty seemed so reasonable. When he returned the digger 15 days late, he was shocked by the penalty fee. What did he pay, and what would he have paid if he had used Company 1 instead? Company 1 Company 2 Day Penalty Day Penalty 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 Which company has a greater 15 day late charge? Describe how the amount of the late charge changes from any given day to the next successive day in both companies 1 and 2. How much would the late charge have been after 20 days under Company 2? At which company does the penalty grow a geometric sequence, at which company does it grow as an arithmetic sequence?
Ex #2: A typical thickness of toilet paper is 0.001 inches. Seems pretty thin, right? Let s see what happens when we start folding toilet paper. How thick is the stack of toilet paper after 1 fold? After 2 folds? After 5 folds? Write an explicit formula for the sequence that models the thickness of the folded toilet paper after n folds. After how many folds will the stack of folded toilet paper pass the 1 foot mark? The moon is about 240,000 miles from Earth. Compare the thickness of the toilet paper folded 50 times to the distance from Earth.
DAY #2 Ex #1: Your long, lost Uncle Nefer recently passed away. Uncle Nefer was always a gambling man and, before he died, won a HUGE lottery jackpot. You were always his favorite so he left you money in his will. Now Uncle Nefer loved math. He LOVED math. So he made his will.interesting. Option #1 Day $ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 You are given two options. Option #1 will give you $10,000 per day, EVERY DAY for an entire month. You even get to pick the month! 1.) Fill in the table showing how much money you will receive each day. 2.) How much money will you receive on the last day of the month? 3.) Look at the $ column. What is the pattern of the numbers? 4.) Would this sequence of numbers be arithmetic or geometric? 5.) Write an explicit formula that models this data. [Hint: an = a1 + (n 1)d] 6.) Use your formula and plug in 31 for n. Do your answers match? 7.) What is the TOTAL amount of money you would receive?
Option #2 Day $ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Option #2 gives you 1 cent on the first day of the month and it doubles every day. 8.) Fill in the table showing how much money you will receive each day. 9.) How much money will you receive on the last day of the month? 10.) Look at the $ column. What is the pattern of the numbers? 11.) Would this sequence of numbers be arithmetic or geometric? 12.) Write an explicit formula that models this data. [Hint: an = a1(r) n 1 ] 13.) Use your formula and plug in 31 for n. Do your answers match? 14.) What is the TOTAL amount of money you would receive?
15.) Which option appears to be modeled by a linear relationship? Explain your reasoning. 16.) Which option appears to be modeled by an exponential relationship? Explain your reasoning. 17.) Which option would you choose? 18.) Isn t exponential growth awesome? So here is our formula that we created for option #2 An =.01(2) n-1 This formula is an example of. Whenever an amount is being repeated multiplied by a decimal, or a percent, or a fraction, or a whole number, it is an example of exponential growth. Ex #1: Suppose you deposited 93 in a bank account getting 2.25% interest. How much money would you have after 10 years? 100 years? 1000 years?
Ex #2: In 2013, a research company found that smartphone shipments (units sold) were up 32.7% worldwide from 2012, with an expectation for the trend to continue. If 959 million units were sold in 2013, how many smartphones can be expected to be sold in 2018 at the same growth rate? (Include the explicit formula for the sequence that models this growth.) Can this trend continue? Explain your reasoning.