Applications of Exponential Equations and Logarithms

Transcription

1 Name: Date: Block: Applications of Exponential Equations and Logarithms DIRECTIONS: Write problem information. Show equations used for solving. Round final answers according to the situation. Label your answers. Interest Applications 1. Chuck s dad puts $2000 in a college fund on his 12 th birthday. The account pays 5% interest. How much does Chuck have in the account when he turns 18? 2. You would like $10,000 for college at age 18. How much would need to be deposited when you were born if the account pays 7.5% interest? 3. Dewitt invested $1500 into an account that pays 9% interest. How long must the money be left in the account for it to grow to $2150? 4. Xavier is 20 years old. He wants to be a millionaire someday. He deposits $10,000 into a savings account with 4% interest. Will the account be enough to make him a millionaire in his lifetime? 5. How many years will it take a sum of money to double if it is invested at 15%? 6. How many years will it take a sum of money to triple if it is invested at 15%? 7. At what annual rate will a sum of money triple in 12 years?

2 8. Tyler buys a new couch from RC Willey for $1549. The store was having a special sale where buyers do not have to start making payments for 2 years, although interest will accrue on the balance. Tyler made no payments for 2 years, but was then surprised that the amount owed had grown to $2549. What interest rate did the store charge? 9. The Rule of 72 can be used to estimate the time it takes for an account to double, t = 72 where r is the interest rate, as a percent (not decimal). a. Use the rule of 72 to estimate the time it takes an account to double if the interest rate is 24%. (use r = 24) r b. Use an exponential equation to find the time it takes to double at 24% (k = 0.24). Compare your answer. Population Applications 10. It is estimated that there are 4100 rabbits on an island, with a growth rate of 55% a year. a. How many rabbits will be on the island in three years? b. At this rate, how long until there are one million rabbits on the island? (Use info from part a) 11. If the world population in 2009 is 7 billion people, and if the population grows continuously at an annual rate of 1.7%, a. What will the population be in 10 years? b. How long will it take to reach 10 billion people?

3 Appreciation and Depreciation 12. The Sawyers bought a condo for $75,000. Assuming that its value will appreciate 6% a year, a. How much will the condo be worth in 5 years? b. How long until the condo is worth $125,000? 13. Jim bought a new car for $18,000. After 4 years, the car was valued at $7500. a. What was the rate of depreciation? b. At this rate of depreciation, how much will the car be worth after 10 years? (Use info from part a) ph Problems 14. The concentration of [H+] is 4.8x10-7, what is the ph? 15. What is the hydrogen ion concentration of a solution of ph 7.4? 16. What is the hydrogen ion concentration of a solution of ph 3.1?

4 17. What is the percent change in [H+] if the ph of a solution, such as blood, changes from ph 7.4 to ph 7.2? 18. What is the percent change in [H+] if the ph of a solution, such as blood changes from ph 7.35 to ph 7.1? Radioactive Decay 19. Radium has a half life of 1620 years. A 20-gram sample is sealed in a box. a. How many grams will be left in 5000 years? b. How long until 7 grams remain? 20. The half life of element X is 57 minutes. Starting with 35 milligrams, a. How many milligrams will be left after 1.5 hours? b. How long will it take to decay to 5 mg? 21. How long does it take for a radioactive element to decay to one eighth of its original mass?

5 22. A radioisotope is used as a power source for a satellite. When launched into orbit, the power output started at 50 watts. The power output decreases at a rate of 0.4% each day. a. Find the power available after 100 days. b. Ten watts of power are required to operate the equipment in the satellite. How long can the satellite continue to operate? 23. To determine the age of fossils, paleontologists measure the amount of Carbon-14 (radiocarbon) remaining the fossil, and compare to that found in living bone tissue. Radiocarbon has a rate of decay of 0.012% per year. A paleontologist finds a bone that might be a dinosaur bone. In the laboratory, she finds that the radiocarbon found in this bone is 1/10 of that found in living bone tissue. If the dinosaurs became extinct 63 million years ago, could this bone have belonged to a dinosaur? 24. Radioactive iodine is used to determine the health of the thyroid gland. It decays at a rate of 8.56% daily. a. Find the half life of this substance. b. Since radioactive iodine decays exponentially, will the substance ever be completely gone? Explain.

6 ANSWERS Applications of Exponential Growth and Decay 1. y =?; n = 2000; k =.05; t = 6 y = 2000e.05(6) y $ y = 10,000; n =?; k =.075; t = = ne.075(18) n $ y = 2150; n = 1500; k =.09; t =? 2150 = 1500e.09t t 4 years 4. y = 1,000,000; n = 10,000; k =.04; t =? = 10000e.04t t 115 years Probably not, unless he lives to be 135 years old. 5. y = 2; n = 1; k =.15; t =? 2 = e.15t t 4.6 years (for y and n, you can use any amount where y is twice as much as n) 6. y = 3; n = 1; k =.15; t =? 3 = e.15t t 7.3 years 7. y = 3; n = 1; k =?; t = 12 3 = e k(12) k 9.2% 8. y = 2549; n = 1549; k =?; t = = 1549e k(2) k 24.9% 9a. Using the formula: t = b. y = 2; n = 1; k = 0.24; t =? 2 = e.24t t 2.9 years 10a. y =?; n = 4100; k =.55; t = 3 y = 4100e.55(3) y 21,000 rabbits 10b. y = 1,000,000; n = 4100; k =.55; t =? = 4100e.55t t 10 years 11a. y =?; n = 7; k =.017; t = 10 y = 7e.017(10) y 8.3 billion people 11b. y = 10; n = 7; k =.017; t =? 10 = 7e.017t t 21 years 12a. y =?; n = 75,000; k =.06; t = 5 y = 75000e.06(5) y $101,000 12b. y = 125,000; n = 75,000; k =.06; t =? = 75000e.06t t 8.5 years About 8.5 years after 1999, so during a. y = 7500; n = 18,000; k =?; t = = 18000e k(4) k 21.9% 13b. y =?; n = 18,000; k.219; t = 10 y = 18000e.219(10) y $ a. y =?; n = 20; k = ln ; t = 5000 y = 20e( ln )5000 y 2.4 grams 14b. y = 7; n = 20; k = ln 2 ; t =? 7 = e( 1620 )t t 2450 years 15a. y =?; n = 35; k = ln 2 ; t = 90 (minutes) 57 y = 35e( 57 )90 y 11.7 mg 15b. y = 5; n = 35; k = ln = 2 35e( ln 57 )t t 160 minutes 1 16., so 3 half-lives a. y =?; n = 50; k =.004; t = 100 y = 50e.004(100) y 33.5 watts 17b. y = 10; n = 50; k =.004; t =? 10 = 50e.004t t 402 days 18. y = 1; n = 10; k =.00012; t =? 1 = 10e.00012t t 19,200 years No, the fossil is less than 20 thousand years old. It could not be a 63-million-year-old dinosaur. 19a. y = 1; n = 2; k =.0856; t =? 1 = 2e.0856t t 8.1 days 19b. Mathematically, the value would never really equal zero. However, since you cannot break up individual atoms, there will eventually be a point where the last atom is gone. 20. ph= 6.3 ln 2 ln 2