To model the temperature inside a building whose initial temperature is 60 F, we

Question

To model the temperature inside a building whose initial temperature is 60 F, we need to solve the equation du/dt = 0.25 (70 - 10*cos (pi*t/12) - u); u(0) = 60. Find u(t)

Explanation / Answer

We have that dT/dt = -k(T - T(a)) from Newtons law of cooling. Here, k is a constant, T is the temperature at time t and T(a) is the ambient temperature....i.e. 5 degrees. We can show that T(t) = T(a) + (T(o) - T(a))e^(-kt) => T(t) = 5 +(T(o) - 5)e^(-kt) so, T(1) = 5 + (T(o) - 5)e^(-k) = 55......i.e. 55 degrees after 1 minute. also, T(5) = 5 + (T(o) - 5)e^(-5k) = 30.....i.e. 30 degrees after 5 minutes. With a little algebraic re-arranging we can deduce that: e^k = (T(o) - 5)/50 and e^5k = (T(o) - 5)/25 => e^4k = 2 => 4k = ln2 i.e. k = (1/4)ln2 = 0.1733 so, e^(0.1733) = (T(o) - 5)/50 => T(o) = 64.5 degrees..........the initial temperature.

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