Solution first find rate constant k given t s 5 t 70

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Solution: First find rate constant k given T S = - 5 , T 0 = 70 , and T (10) = 60 . T - T S = ( T 0 - T S ) e kt T (10) - ( - 5) = (70 - ( - 5)) e k (10) 60 + 5 = 75 e 10 k 65 = 75 e 10 k 65 75 = e 10 k ln 65 75 = ln e 10 k = 10 k k = ln(65 / 75) 10 ≈ - 0 . 0143 .
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Then find time t when T = 40 and T 0 = 60 . T - T S = ( T 0 - T S ) e kt 40 + 5 = (60 - ( - 5)) e kt 45 65 = e kt ln 45 65 = kt t = ln(45 / 65) k = 10 ln(45 / 65) ln(65 / 75) 25 . 7 min Example 4 Continuously Compounded Interest. Find time t given interest rate r and an amount relative to A 0 . How long will it take an investment to increase in value by 50% if the interest rate is 4% compounded continuously? Solution: We wish to find t when A ( t ) = 1 . 5 A 0 given interest rate r = 0 . 04 . A ( t ) = A 0 e rt 1 . 5 A 0 = A 0 e 0 . 04 t 1 . 5 = e 0 . 04 t ln 1 . 5 = 0 . 04 t t = ln 1 . 5 0 . 04 10 . 14 years S. Chang CU-Boulder Dept of Applied Mathematics
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