507 4.6 Applications and Models of Exponential Growth and Decay 4.6 Exercises CONCEPT PREVIEW Population Growth A population is increasing according to the exponential function y = 2e0.02t, where y is in millions and t is the number of years. Match each question in Column I with the correct procedure in Column II to answer the question. I 1. How long will it take for the population to triple? 2. When will the population reach 3 million? 3. How large will the population be in 3 yr? 4. How large will the population be in 4 months? II A. Evaluate y = 2e0.0211/32. B. Solve 2e0.02t = 6. C. Evaluate y = 2e0.02132. D. Solve 2e0.02t = 3. CONCEPT PREVIEW Radioactive Decay Strontium-90 decays according to the exponential function y = y0 e-0.0241t, where t is time in years. Match each question in Column I with the correct procedure in Column II to answer the question. I 5. If the initial amount of Strontium-90 is 200 g, how much will remain after 10 yr? 6. If the initial amount of Strontium-90 is 200 g, how much will remain after 20 yr? 7. What is the half-life of Strontium-90? 8. How long will it take for any amount of Strontium-90 to decay to 75% of its initial amount? II A. Solve 0.75y0 = y0 e-0.0241t. B. Evaluate y = 200e-0.02411102. C. Solve 1 2 y0 = y0 e-0.0241t . D. Evaluate y = 200e-0.02411202. (Modeling) Exercises are grouped according to discipline. See Examples 1–6. Physical Sciences (Exercises 9–28) An initial amount of a radioactive substance y0 is given, along with information about the amount remaining after a given time t. For an equation of the form y = y0 ekt that models the situation, give the exact value of k in terms of natural logarithms. 9. y0 = 60 g; After 3 hr, 20 g remain. 10. y0 = 30 g; After 6 hr, 10 g remain. 11. y0 = 10 mg; The half-life is 100 days. 12. y0 = 20 mg; The half-life is 200 days. 13. y0 = 2.4 lb; After 2 yr, 0.6 lb remains. 14. y0 = 8.1 kg; After 4 yr, 0.9 kg remains. Solve each problem. 15. Decay of Lead A sample of 500 g of radioactive lead-210 decays to polonium-210 according to the function A1t2 = 500e-0.032t, where t is time in years. Find the amount, to the nearest gram, of radioactive lead remaining after (a) 4 yr, (b) 8 yr, (c) 20 yr. (d) Find the half-life to the nearest year.
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