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Pre-Calc Homework Solutions 286

# Pre-Calc Homework Solutions 286 - 286 Chapter 6 Review 47 T...

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43. continued (b) Note that we choose a small interval of x -values because the y -values decrease very rapidly and our calculator cannot handle the calculations for x # 2 1. (This occurs because the analytic solution is y 5 2 2 1 ln (2 2 e 2 x ), which has an asymptote at x 5 2 ln 2 < 2 0.69. Obviously, the Euler approximations are misleading for x # 2 0.7.) [ 2 1, 0.2] by [ 2 10, 2] 44. Set y 1 5 2 } x e 2 y 1 1 x y } and use IMPEULG with initial values x 5 0 and y 5 0 and step sizes 0.1 and 2 0.1. (a) [ 2 0.2, 4.5] by [ 2 5, 1] (b) [ 2 4.5, 0.2] by [ 2 1, 5] 45. (a) Half-life 5 } ln k 2 } 2.645 5 } ln k 2 } k 5 } 2 l . n 64 2 5 } < 0.262059 (b) Mean life 5 } 1 k } < 3.81593 years 46. T 2 T s 5 ( T 0 2 T s ) e 2 kt T 2 40 5 (220 2 40) e 2 kt Use the fact that T 5 180 and t 5 15 to find k . 180 2 40 5 (220 2 40) e 2 ( k )(15) e 15 k 5 } 1 1 8 4 0 0 } 5 } 9 7 } k 5 } 1 1 5 } ln } 9 7 } T 2 40 5 (220 2 40) e 2 ((1/15) ln (9/7)) t 70 2 40 5 (220 2 40) e 2 ((1/15) ln (9/7)) t e ((1/15) ln (9/7)) t 5 } 1 3 8 0 0 } 5 6 1 } 1 1 5 } ln } 9 7 } 2 t 5 ln 6 t 5 } l 1 n 5 ( l 9 n /7 6 ) } < 107 min It took a total of about 107 minutes to cool from 220 8 F to 70 8 F. Therefore, the time to cool from 180 8 F to 70 8 F was about 92 minutes. 47. T 2 T s 5 ( T 0 2 T s ) e 2 kt We have the system: 5 Thus, } 4 3 6 9 2 2 T T s s } 5 e 2 10 k and } 3 4 3 6 2 2 T T s s } 5 e 2 20 k . Since ( e 2 10 k ) 2 5 e 2 20 k , this means: ( } 4 3 6 9 2 2 T T s s } ) 2 5 } 3 4 3 6 2 2 T T s s }
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