MarcKrishkeHandout5.6 - Math 1540 Handout: Growth and Decay...

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Math 1540 Handout: Growth and Decay 1.) The population of the United States in 2000 was 282 million people. In addition, the population of the U.S. was growing at a rate of 1.1% per year. Assuming that this growth rate continues, the model 2000 ( ) 282(1.011) t P t - = represents the population P (in millions of people) in year t . a.) When will the population of the U.S. be 303 million people? 2000 2000 2000 ( ) 303 282(1.011) 303 (1.011) 1.074468 ln(1.011) ln1.074468 t t t P t - - - = = = = ( 2000)ln1.011 ln1.074468 ln1.074468 2000 ln1.011 t t - = - = 2000 6.565452 2006.565452 t t - = = Answer: In the year 2006. b.) When will the population of the U.S. be 355 million people? 2000 2000 2000 ( ) 355 282(1.011) 355 (1.011) 1.258865 ln(1.011) ln1.258865 t t t P t - - - = = = = ( 2000)ln1.011 ln1.258865 ln1.258865 2000 ln1.011 t t - = - = 2000 21.043143 2021.043143 t t - = = Answer: In the year 2021.
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2.) The value V of a Chevy Cobalt that is t years old can be modeled by ( ) 14,512(0.82) t V t = . a.) According to the model, when will the car be worth $9000? 9000 14512(0.82) 0.6201764 (0.82) ln0.6201764 ln(0.82) ln0.6201764 ln0.82 ln0.6201764 ln0.82 2.4074 t t t t t t = = = = = = b.) When will the car be worth $4,000? 4000 14512(0.82) 0.27563396 (0.82) ln0.27563396 ln(0.82) ln0.27563396 ln0.82 ln0.27563396 ln0.82 6.4937 t t t t t t = = = = = = c.) When will the car be worth $2,000? 2000 14512(0.82) 0.13781698 (0.82) ln0.13781698 ln(0.82) ln0.13781698 ln0.82 ln0.13781698 ln0.82 9.98649 t t t t t t = = = = = = 3.) A colony of bacteria increases according to the law of exponential growth. a.) If the number of bacteria doubles in 3 hours, find the function that gives the number of cells in the culture. ( ) kt o A t A e = when t = 3 hours A (t) = 2 A o : 3 2 k o o A A e =
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3 2 k e = 3 ln2 ln k e = ln2 3 ln ln2 3 ln2 3 k e k k = = = Then 3 ln 2 1 ln 2 ln 2 3 3 3 ( ) 2 t t t t o o o o A t A e A e A e A = = = = Thus, 3 ( ) 2 t o A t A = b.) How long will it take for the size of the colony to triple? Substitute 3
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MarcKrishkeHandout5.6 - Math 1540 Handout: Growth and Decay...

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