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Unformatted text preview: Math 107 Third Midterm Examination November 22, 2005 NAME (Please print) Page Score 2 3 4 5 6 7 Total (Max Possible: 100) Instructions: 1. Do all computations on the examination paper. You may use the backs of the pages if necessary. 2. Put answers inside the boxes (when applicable). 3. Please signify your adherence to the honor code: I, , have neither given nor received aid in completion of this examination. 1 (10 Points) Score A can of orange juice concentrate is taken from a freezer at- 5 into a room with temperature 20 degrees. After 30 minutes, the temperature of the orange juice con- centrate is- 2 degrees. Newtons Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). 1. (3 points) Describe the variables you will use to solve this problem, including their units. Let t be time in minutes, T ( t ) be temperature in degrees and k be the proportionality constant in Newtons law, with units of 1 /minutes . 85% got this mostly right. 2. (3 points) Describe your model for this problem, including a differential equation and one or more conditions on the solution. Newtons law says dT dt ( t ) = k [ T ( t )- 20]; the initial condition is T (0) =- 5 and the other condition is T (30) =- 2. 90% got this mostly right. 3. (4 points) How long will it take for the juice to reach degrees? The homogeneous solution is T H ( t ) = c 1 e kt and the particular solution is T P ( t ) = c 2 . Plugging T P into the differential equation gives us c 2 = 20. The initial condition implies that- 5 = T (0) = c 1 + 20 = c 1 =- 25 The other condition determines k :- 2 = T (30) =- 25 e 30 k + 20 = e 30 k = 22 / 25 = k = 1 30 ln 22 25 Now we solve for the time at which T ( t ) = 0: 0 = T ( t ) =- 25 e kt + 20 = t = 1 k ln 20...
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