105-Mid2Prac2Sols

105-Mid2Prac2Sols - Math 105 Practice Midterm 2 for Midterm...

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Math 105 - Practice Midterm 2 for Midterm 2 Solutions This practice midterm may be harder and/or longer than the real midterm. Not all question will be worth the same number of points. 1. Find the area of the region in the first quadrant that is bounded above by y = x and below by the x -axis and the line y = x - 2 . See the last page of these solutions. 2. A bank account has $20 , 000 earning 5% interest compounded continuously. A pensioner uses the account to pay himself an annuity, drawing continuously at a $2000 annual rate. How long will it take for the balance in the account to drop to zero? dx dt = 0 . 05 x - 2000 = 0 . 05( x - 40000) , x (0) = 20000 Z 1 x - 40000 dx = Z 0 . 05 dt ln | x - 40000 | = 0 . 05 t + C 1 . ± ( x - 40000) = e 0 . 05 t + C 1 = e C 1 e 0 . 05 t x = 40000 + C 2 e 0 . 05 t , where C 2 can be any number ( e C 1 can only be positive, but because of the ± we can have C 2 = ± e C 1 ). 20000 = x (0) = 40000 + C 2 e 0 . 05 · 0 = 40000 + C 2 C 2 = - 20000 x ( t ) = 40000 - 20000 e 0 . 05 t = 20000(2 - e 0 . 05 t ) . The balance will equal zero when 0 = 2 - e 0 . 05 t t = 1 0 . 05 ln(2) = 20 ln(2) ( 14 years) .
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