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Math 207

Written Homework 4

2022

Problems

1.   A recently abandoned building has an interior temperature of 70 ∘ F  at midnight.  It has a time constant of 2 hours.  The outdoor temperature fluctuates sinusoidally,    reaching a maxmimum of 60 ∘ F at noon and a minimum of 20∘ F at midnight every   day.  Assume there are no heating or cooling effects on the building aside from        Newton’s Law of Cooling.

a.   [3 pts] Write a function modeling the outdoor temperature as a function of time.

b.   [3 pts] Use your answer to part a to model the above situation as an initial value problem.

c.    [4 pts] Solve the initial value problem from part b. (Feel free to use a table of integrals.  But make sure to justify and clearly show your work otherwise)

d.   [2 pts] The foundation of the building will be damaged if the interior                     temperature ever reaches the freezing point 32∘ F.  Will that ever happen? (Feel free to answer this by examining the graph.)

2.   [8 pts] On an alien planet (with unknown gravity), a 5 kilogram object is dropped      from rest 10 meters above the ground. In the atmosphere of this planet, the object is estimated to have a drag constant of −3 kg/sec. (Assume drag is proportional to        velocity) The object takes 15 seconds to hit the ground. Use this information to          calculate the acceleration due to gravity on this planet.