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math10bSolFinal - Math 10B Final Exam Solutions 150 pts...

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Math 10B Final Exam Solutions 150 pts December 8, 2009 Professor Evans Directions: Answers alone are not sufficient. Justify, and show all work. The multiple part problems 4,6,7 are worth 18 points each. The other six problems are worth 16 points each. (1) A man was murdered in a 34.6 degree Fahrenheit meat locker. At noon, the corpse temper- ature was 38.6, and an hour later, it had dropped to 36.6. The man was murdered at (a) 6 am (b) 7 am (c) 8 am (d) 9 am (e) none of these times Solution : Let T ( t ) denote the temperature of the corpse at time t , with noon being chosen as the “zero hour” t = 0. The solution to the differential equation dT/dt = k (34 . 6 - T ) is T ( t ) = 34 . 6 + Be - kt . Plug in t = 0 to get 38 . 6 = 34 . 6 + B , so B = 4 and T = 34 . 6 + 4 e - kt . Plug in t = 1 to get 36 . 6 = 34 . 6 + 4 e - k . Thus e - k = 1 / 2 and T = 34 . 6 + 4(1 / 2) t . At the time of the murder, 98 . 6 = 34 . 6 + 4(1 / 2) t , so 16 = (1 / 2) t . Thus t = - 4, so the murder took place at 8 am. (2) An upright vase of height 16 is formed by rotating the curve y = x 4 (0 x 2) around the y -axis. Using an integral, find the volume of water that it takes to fill the vase.
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