Consider an insulated box (a building, perhaps) with internal temperature u(t).

Consider an insulated box (a building, perhaps) with internal temperature u(t). According

to Newton’s law of cooling, u satisfies the differential equation

du/dt= −k[u − T(t)], (i)

where T(t) is the ambient (external) temperature. Suppose that T(t) varies sinusoidally;

for example, assume that T(t) = T0 + T1 cos ωt.

(a) Solve Eq. (i) and express u(t) in terms of t, k, T0, T1, and ω. Observe that part of

your solution approaches zero as t becomes large; this is called the transient part. The

remainder of the solution is called the steady state; denote it by S(t).

(b) Suppose that t is measured in hours and that ω = π/12, corresponding a period of 24 h

for T(t). Further, let T0 = 60◦F, T1 = 15◦F, and k = 0.2/h. Draw graphs of S(t) and T(t)

versus t on the same axes. From your graph estimate the amplitude R of the oscillatory

part of S(t). Also estimate the time lag τ between corresponding maxima of T(t) and S(t).

(c) Let k, T0, T1, and ω now be unspecified. Write the oscillatory part of S(t) in the form

Rcos[ω(t − τ)]. Use trigonometric identities to find expressions for R and τ . Let T1 and

ω have the values given in part (b), and plot graphs of R and τ versus k.