7-8 Exponential Growth and Decay Notes

Transcription

1 7-8 Eponential Growth and Decay Notes Decay y = a b where a > 0 and b is between 0 and 1 Eample : y = 100 (.5) As is increases by 1, y decreases to 1/2 of its previous value. Growth y = a b where a > 0 and b is greater than 1 Eample : y = 1 (2) As is increases by 1, y increases by a factor of 2 (doubled). Any quantity that grows or decays by a fied percent at regular intervals is said to possess eponential growth or eponential decay. When a quantity grows by a fied percent at regular intervals in an applied situation, the pattern can be represented by the functions, Growth: Decay: Growth Eample Banking: A bank account balance, b, for an account starting with s dollars, earning an annual interest rate, r, and left untouched for n years can be calculated as b = s(1 + r) n (an eponential growth formula). Find a bank account balance to the nearest dollar, if the account starts with $100, has an annual rate of 4%, and the money left in the account for 12 years.

2 Growth Eample: In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after How many cell phone subscribers were in Centerville in 1994 (9 years later)? (Don't consider a fractional part of a person.) Growth By Doubling: One of the most common eamples of eponential growth deals with bacteria. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. If we start with only one bacteria which can double every hour, by the end of one day how many bacteria will we have? (Note t = total time/doubling time) Decay by one-half: Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds? Half-Life Decay Eample: The half-life of a substance is the amount of time it takes for half of the amount of a substance to decay. If the half-life of the pesticide DDT is 15 years (1 half life=15 years), how many years will it take for 100 grams of DDT to decay to 5 grams? Note: t is the number of half-lives. To find the total years, you multiply the number of half-lives by the number of years for one half-life to occur.

3 Name: Date: Period: Unit 7-8 Eponential Growth and Decay Homework 1. A $22,000 truck depreciates 11% per year. Write an eponential function to model the situation. Find the value of the function after 8 years. Round to the nearest cent. 2. A population of 2785 brown bears increases 3% each year. Write an eponential function to model each situation. Find the value of the function after 8 years. Round to nearest integer (whole bear). 3. If you invest $2000 compounded annually at 6%, how long will it take to double your investment? Round to 3 decimal places. 4. Suppose you deposit $1000 in an account paying 3% annual interest, compounded continuously. r t Using A = Pe, find the balance after 10 years. Round to the nearest cent. 5. Suppose your savings account balance is currently $5,200 in an account paying 5% annual interest, r t compounded continuously. Using A = Pe, calculate how much money you deposited when you first opened the account 18 years ago. Round to the nearest cent.

4 6. (a) A bacteria doubles every 7 hours. The biologist has 48 bacteria now. How many will there be in 42 hours? (b) When will there be 1200 bacteria remaining? Round to 3 decimal places. 7. A chemical substance has a half-life of 5 hours. (a) How much of a 12-gram sample remains after 12 hours? Round to 3 decimal places. (b) When will there be only 3 grams remaining? 8. Jeanine bought a car 8 years ago for $18,000. The car is now worth $3,500. Assuming a steady rate of depreciation, use A = a(1 r) t to find the yearly rate of depreciation (epress as a percent with 1 decimal place).

5 Answers: 1. $8, bears years 4. $1, $2, a) 3072 bacteria b) hours 7. a) grams b) 10 hours % depreciation per year

6 Formula Sheet for Growth & Decay Compounded Continuously: A = Pe r t Eponential Growth: A = a(1 + r) t Eponential Decay: A = a(1 r) t Half-Life: t 1 k A = a ; k = half life 2 t A = a 2 k ; k = doubling time Doubling: ( )