nsol2 - Solutions to Sample Problems for Exam Two A...

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Solutions to Sample Problems for Exam Two A. APPLICATIONS: Write a differential equation which describes the following physical problems and then solve. Give the limiting value of the solution, if any. In general, equation x = ax + b has solution x = - b a + Ce at . When a < 0, the limiting value of x is L = - b/a . 1. Orange juice at 30 cools to 20 in 10 minutes in a 5 refrigerator. Find the equation Θ( t ) for the temperature at time t. Θ = - k - 5) = - k Θ + 5 k . Solution: Θ = 5 + Ce - kt . 30 = Θ(0) = 5 + C , so C = 25. 20 = Θ(10) = 5 + 25 e - 10 k , so e - 10 k = 20 - 5 2 5 = 3 / 5, - 10 k = ln (3 / 5), k = ln (5 / 3) / 10 and e - k = 3 5 1 / 10 . Then Θ = 5 + 25 e ln (3 / 5) t 10 = 5 + 25(3 / 5) t/ 10 . The limiting value of Θ is 5. 2. A student consumes 2 ounces per minute of coffee, containing 10 mg per ounce of caffiene. Assuming that the total liquid in his body remains at 800 ounces, find the equation for the amount C(t) of caffiene in his body at time t. Estimate the concentration after 60 minutes. Let A ( t ) be the amount of caffiene, so C ( t ) = A ( t ) / 800. A = 20 - A/ 400, so a = - 1 / 400, b = 20 and - b/a = 8000. A = 8000 + ke - t/ 400 . 0 = A (0) = 8000 + k , so k = - 8000. A = 8000 - 8000 e - t/ 400 and C = A/ 800 = 10 - 10 e - t/ 400 . C (60) = 10 - 10 e - 60 / 400 = 1 . 4. The limiting value of A is 8000 and the limiting value of C is 10. 3. A graduate takes out a new car loan for $5000 at 6% (i.e. 1/2% monthly) with monthly payments of $100. Find a formula for the amount P(t) of principle outstanding after t months and estimate the life of the loan. P = . 005 P - 100, so a = . 005, b = - 100 and - b/a = 20000. P = 20000 + Ce . 005 t 5000 = P (0) = 20000 + C , so C = - 15000. P = 20000 - 15000 e . 005 t . Solve 0 = P ( t ) to get e . 005 t = 4 / 3, so t = 200 ln (4 / 3) = 57 . 5 months. There is no limiting value because a = . 005 > 0. 1
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2 4. A calculus book subject to air resistance of magnitude . 2 mv falls from a height of 4000 feet. Write and solve the differential equation for the velocity v , find the equation of motion x ( t ), limiting velocity and approximate time of impact.
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