Question

Suppose that your air conditioner fails on Sunday at midnight (t0 = 0), and you cannot afford to have it repaired until payday at the end of the month. Assume that the outside temperature varies according to the function.

A(t)= 80 − 5 cos(π/12t)-5√3sin(π/12t)

and that your inside temperature, u(t) obeys Newton’s law of cooling and is governed by the differential equation

du/dt=−0.2(u − A(t))

(a) If your indoor temperature when the air conditioner failed was 70◦F, determine the dynamics of temperature inside your apart- ment over time. i.e. find a particular solution to the initial value problem.

(b) What will the temperature inside the apartment be, 24 hours after the break down?

(c) In the long run (t → ∞), what is the maximum and minimum temperature you can anticipate inside your apartment?

(d) Plot the graph of the outdoor and indoor temperature on the same axis and comment on how long it takes for the indoor temperature to reach a maximum after the outdoor temperature peaks.

Answer 1