14. 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 = -k(u - T(t)), dt (37) where T(t) is the ambient (external) temperature. Suppose that T(t) varies sinusoidally; for example, assume that T(t) = To+T₁ cos(wt). a. Solve equation (37) and express u(t) in terms of t,k, To, T₁, and w. 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). Gb. Suppose that t is measured in hours and that wπ/12, corresponding to a period of 24 h for T(t). Further, let To = 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 7 between corresponding maxima of T(t) and S(t). c. Let k, To, T₁, and w now be unspecified. Write the oscillatory part of s(t) in the form R cos(w(t)). Use trigonometric identities to find expressions for R and 7. Let T₁ and w have the values given in part b, and plot graphs of R and 7 versus k.