5_8 - Chapter 5 Exponential and Logarithmic Functions 5.8...

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Chapter 5 727 Exponential and Logarithmic Functions 5.8 Exponential Growth and Decay; Newton’s Law; Logistic Models 1. P ( t ) = 500 e 0.02 t (a) P (0) = 500 e 0.02 ( 29 ⋅ 0 ( 29 = 500 flies (b) growth rate = 2 % (c) P (10) = 500 e 0.02 ( 29 ⋅ 10 ( 29 = 611 flies (d) Find t when P = 800: 800 = 500 e 0.02 t 1.6 = e 0.02 t ln1.6 = 0.02 t t = ln1.6 0.02 23.5 days (e) Find t when P = 1000: 1000 = 500 e 0.02 t 2 = e 0.02 t ln2 = 0.02 t t = ln2 0.02 34.7 days 2. N ( t ) = 1000 e 0.01 t (a) N (0) = 1000 e 0.01 ( 29 ⋅ 0 ( 29 = 1000 bacteria (b) growth rate = 1 % (c) N (4) = 1000 e 0.01 ( 29 ⋅ 4 ( 29 = 1041 bacteria (d) Find t when N = 1700: 1700 = 1000 e 0.01 t 1.7 = e 0.01 t ln1.7 = 0.01 t t = ln1.7 0.01 53.1 hours (e) Find t when N = 2000: 2000 = 1000 e 0.01 t 2 = e 0.01 t ln2 = 0.01 t t = ln2 0.01 69.3 hours
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Chapter 5 Exponential and Logarithmic Functions 728 3. A ( t ) = A 0 e - 0.0244 t = 500 e - 0.0244 t (a) decay rate = 2.44 % (b) A (10) = 500 e - 0.0244 ( 29 10 ( 29 391.74 grams (c) Find t when A = 400: 400 = 500 e - 0.0244 t 0.8 = e - 0.0244 t ln0.8 = - 0.0244 t t = ln0.8 - 0.0244 9.15 years (d) Find t when A = 250: 250 = 500 e - 0.0244 t 0.5 = e - 0.0244 t ln0.5 = - 0.0244 t t = ln0.5 - 0.0244 28.4 years 4. A ( t ) = A 0 e - 0.087 t = 100 e - 0.087 t (a) decay rate = 8.7 % (b) A (9) = 100 e - 0.087 ( 29 9 ( 29 45.70 grams (c) Find t when A = 70: 70 = 100 e - 0.087 t 0.7 = e - 0.087 t ln0.7 = - 0.087 t t = ln0.7 - 0.087 4.10 days (d) Find t when A = 50: 50 = 100 e - 0.087 t 0.5 = e - 0.087 t ln0.5 = - 0.087 t t = ln0.5 - 0.087 7.97 days 5. Use N ( t ) = N 0 e kt and solve for k : 1800 = 1000 e k (1) 1.8 = e k k = ln1.8 When t = 3: N (3) = 1000 e ln1.8 ( 29 3 ( 29 = 5832 mosquitos Find t when N ( t ) = 10,000 : 10,000 = 1000 e ln1.8 ( 29 t 10 = e ln1.8 ( 29 t ln10 = ln1.8 ( 29 t t = ln10 ln1.8 3.9 days 6. Use N ( t ) = N 0 e kt and solve for k : 800 = 500 e k (1) 1.6 = e k k = ln1.6 When t = 5 : N (5) = 500 e ln1.6 ( 29 5 ( 29 = 5243 bacteria Find t when N ( t ) = 20,000 : 20,000 = 500 e ln1.6 ( 29 t 40 = e ln1.6 ( 29 t ln40 = ln1.6 ( 29 t t = ln40 ln1.6 7.85 hours
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Section 5.8 Exponential Growth and Decay; Newton’s Law; Logistic Models 729 7. Use P ( t ) = P 0 e kt and solve for k : 2 P 0 = P 0 e k (1.5) 2 = e 1.5 k ln2 = 1.5 k k = ln2 1.5 When t = 2 : P (2) = 10,000 e ln2 1.5 2 ( 29 = 25,198 is the population 2 years from now. 8.
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This homework help was uploaded on 04/07/2008 for the course MAC 1105 taught by Professor Any during the Spring '08 term at FIU.

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5_8 - Chapter 5 Exponential and Logarithmic Functions 5.8...

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