HPC Review of Exponents Day 5

HPC Review of Exponents [Day 5] Name: Exponential Function: Exponential Growth: 1. Exponential Decay: Domain: Domain: Range: Range: Complete the table below. The cost of each item grows exponentially. Item Annual Rate of Increase Cost Now Cost in 𝟏𝟎 years Cost in 𝟐𝟎 years a. Airplane Ticket 15% $300 b. Jar of Mustard 7% $1 c. College Tuition 10% $12000 2. Complete the table below. The value of each item decays exponentially. Item a. Farm Tractor b. Value of the c. 3. 4. Dollar Value of the Dollar Annual Rate of Decrease 25% 6% Cost Now $65000 $1 8% $1 Value in 𝟑 years The value of a car in dollars 𝑡 years from now is 𝑉(𝑡) = 12500(0.85)𝑡 . a. What is the annual rate of depreciation (the rate at which the car loses value?) b. In how many years will the value of the car be approximately half what it is now? The value of a car in dollars 𝑡 years from now is 𝑉(𝑡) = 4000(0.85)𝑡 . a. What is the annual rate of depreciation (the rate at which the car loses value?) b. In how many years will the value of the car be approximately half what it is now? Value in 𝟔 years 5. The cost of a certain brand of camera has been increasing at 8% per year. If a camera now costs $150, find the cost: a. 6. 2 years and 6 months from now b. 4 years and 3 months ago The value of a computer depreciates at the rate of 25% per year. If a computer is now worth $2400, find its approximate value: a. 3 years and 6 months from now b. 20 months ago 7. Suppose the population of a nation grows at 3% per year. If the population was 30,000,000 people in 1990, what will be the population, to the nearest million, in the year 2000? 8. If $1000 is invested so that it grows at the rate of 10% per year, what will the investment be worth in 20 years? Half-Life: 𝑡 9. 10. 11. A colony of bacteria decays so that the population 𝑡 days from now is given by (𝑡) = a. What is the amount present when 𝑡 = 0? b. How much will be present in 4 days? c. What is the half-life? 1 4 1000 (2) . The half-life of a radioactive isotope is 4 days. If 3.2 kg are present now, how much will be present after: a. 4 days? b. 8 days? c. 20 days d. 𝑡 days? The half-life of radium is about 1600 years. If 1 kg is present now, how much will be present after: a. 3200 years? b. 16000 years? c. 800 years d. 𝑡 years? The table shows the amount 𝐴(𝑡) in grams of a radioactive element present after 𝑡 days. Suppose that 𝐴(𝑡) decays exponentially. 12. 𝒕 days 𝑨(𝒕) 0 320 2 226 4 160 a. What is the half-life of the element? b. About how much will be present after 16 days? c. Find an equation for 𝐴(𝑡). 6 115 8 80 10 57 13. When a certain medicine enters the bloodstream, it gradually dilutes, decreasing exponentially with a half-life of 3 days. The initial amount of the medicine in the bloodstream is 𝐴0 milliliters. What will the amount be 30 days later? 14. An amount 𝐴0 of a radioactive iodine has a half-life of 8.1 days. In terms of 𝐴0 , how much is present after 5 days? 15. A bacteria colony triples every 6 days. The population is 𝑃0 bacteria. What will the population 𝑃(𝑡) be 𝑡 days later? 16. Find an exponential function having the given values. a. 𝑓(0) = 3, 𝑓(1) = 15 b. 𝑓(0) = 5, 𝑓(3) = 40 c. 𝑓(0) = 64, 𝑓(2) = 4 d. 𝑓(0) = 80, 𝑓(4) = 5 18. 𝑥 3 − 6𝑥 3 + 8 = 0 Solve 2 17. 1 𝑥 3 − 7𝑥 3 + 12 = 0 4 2 Factor 𝑥 2𝑦3 − 𝑥 4𝑦2 19. 3 2 1 2 1 2 20. 3 2 21. 𝑥 𝑦 −𝑥 𝑦 23. (𝑥 − 1)2 − 𝑥(𝑥 − 1)− 2 25. (𝑥 2 + 1)2 − 𝑥 2 (𝑥 2 + 1)2 1 1 2 − 1 2 3 2 −𝑥 𝑦 1 2 22. 𝑥 𝑦 24. (𝑥 + 1)2 − 4(𝑥 + 1)2 26. (𝑥 2 + 2)2 − 𝑥 2 (𝑥 2 + 2)− 2 1 3 𝑥 5𝑦2 − 𝑥 3𝑦7 3 1 1 1 1 1a. $1214; $4910 1b. $1.97; $3.87 1c. $31125; $80730 2a. $27422; $11569 2b. $0.83; $0.69 2c. $. 078; $0.61 3a. 15% 3b. ≈ 4 years 4a. 15% 4b. ≈ 4 years 5a. $182 5b. $108 6a. $877 6b. $3877 7. 40 million people 8. $6727 9a. 1000 9b. 500 10a. 1.6 kg 10b. 0.8 kg 10c. 0.1 kg 10d. 3.2 (2) 11a. 250 g 12a. 4 days 12b. 20 g 11b. 0.98 g 𝑡 12c. 16a. 𝑓(𝑥) = 3(5)𝑥 1 0.617 𝑡 15. 𝑃0 (3)6 13. 𝐴0 (2) 14. 𝐴0 (2) 16b. 𝑓(𝑥) = 5(2)𝑥 16c. 𝑓(𝑥) = 64 (4) 18. 2 , 3 2 22. 𝑥 2 𝑦 − 2 (1 − 𝑥𝑦) 25. (𝑥 2 + 1)2 19. 1 𝑡 1600 11d. (2) 1 10 1 4 320 (2) 1 1 11c. 707 g 1 1 𝑥 𝑥 2 𝑦 2 (𝑦 − 𝑥 2 ) 20. 1 1 26. 𝑥 3 𝑦 2 (𝑥 2 − 𝑦 5 ) 1 𝑥 16d. 𝑓(𝑥) = 80 (2) 1 2(𝑥2 + 2) 1 −2 −(𝑥 − 1)− 2 17. 27, 64 1 21. 𝑥 2 𝑦 2 (𝑥 − 𝑦 ) 24. (𝑥 + 1)2 (𝑥 − 3) 1 23. 9c. 4 days 𝑡 4 1