SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 363 Applications and Extensions 5.8 Assess Your Understanding 1. Growth of an Insect Population The size P of a certain insect population at time t (in days) obeys the law of uninhibited growth ( ) = P t e 500 .t 0.02 (a) Determine the number of insects at = t 0 days. (b) What is the growth rate of the insect population? (c) Graph the function using a graphing utility. (d) What is the population after 10 days? (e) When will the insect population reach 800? (f) When will the insect population double? 2. Growth of Bacteria The number N of bacteria present in a culture at time t (in hours) obeys the law of uninhibited growth ( ) = N t e 1000 .t 0.01 (a) Determine the number of bacteria at = t 0 hours. (b) What is the growth rate of the bacteria? (c) Graph the function using a graphing utility. (d) What is the population after 4 hours? (e) When will the number of bacteria reach 1700? (f) When will the number of bacteria double? 3. Radioactive Decay Strontium-90 is a radioactive material that decays according to the function ( ) = − A t A e ,t 0 0.0244 where A0 is the initial amount present and A is the amount present at time t (in years). Assume that a scientist has a sample of 500 grams of strontium-90. (a) What is the decay rate of strontium-90? (b) Graph the function using a graphing utility. (c) How much strontium-90 is left after 10 years? (d) When will 400 grams of strontium-90 be left? (e) What is the half-life of strontium-90? 4. Radioactive Decay Iodine-131 is a radioactive material that decays according to the function ( ) = − A t A e ,t 0 0.087 where A0 is the initial amount present and A is the amount present at time t (in days). Assume that a scientist has a sample of 100 grams of iodine-131. (a) What is the decay rate of iodine-131? (b) Graph the function using a graphing utility. (c) How much iodine-131 is left after 9 days? (d) When will 70 grams of iodine-131 be left? (e) What is the half-life of iodine-131? 5. Growth of a Colony of Mosquitoes The population of a colony of mosquitoes obeys the law of uninhibited growth. (a) If N is the population of the colony and t is the time in days, express N as a function of t . (b) If there are 1000 mosquitoes initially and there are 1800 after 1 day, what is the size of the colony after 3 days? (c) How long is it until there are 10,000 mosquitoes? 6. Bacterial Growth A culture of bacteria obeys the law of uninhibited growth. (a) If N is the number of bacteria in the culture and t is the time in hours, express N as a function of t . (b) If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the culture after 5 hours? (c) How long is it until there are 20,000 bacteria? 7. Population Growth The population of a southern city is growing according to the exponential law. (a) If N is the population of the city and t is the time in years, express N as a function of t . (b) If the population doubled in size over an 18-month period and the current population is 10,000, what will the population be 2 years from now? 8. Population Decline The population of a midwestern city is declining according to the exponential law. (a) If N is the population of the city and t is the time in years, express N as a function of t . (b) If the population decreased from 900,000 to 800,000 from 2016 to 2018, what will the population be in 2020? 9. Radioactive Decay The half-life of radium is 1690 years. If 10 grams is present now, how much will be present in 50 years? (c) Solve the equation ( ) = P t 50. ( ) ( ) + = = + ≈ + ≈ ≈ ≈ ≈ e e e e e t t 100.3952 1 0.0316 50 100.3952 50 1 0.0316 2.0079 1 0.0316 1.0079 0.0316 31.8956 ln 31.8956 0.0581 59.6 years t t t t t 0.0581 0.0581 0.0581 0.0581 0.0581 Divide both sides by 50. Subtract 1 from both sides. Divide both sides by 0.0316. Rewrite as a logarithmic expression. Divide both sides by 0.0581. It will take approximately 59.6 years for the percentage of long-life-span wood products remaining to reach 50%. (d) The numerator of 100.3952 is reasonable because the maximum percentage of wood products remaining that is possible is 100%. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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