Section 4.3– Number 38 The temperature, T, in oC, of a yam put into a 200oC oven is given as a function of time, t, in minutes, by T = a(1 – e−kt) + b a) If the yam starts at 20oC, find a and b. Solution: Since the yam starts at 20oC, we have that T = 20 when t = 0. Substituting into the formula, we get 20 = a(1 – e−k(0)) + b 20 = a(1 – e0) + b 20 = a(1 – 1) + b 20 = a(0) + b 20 = b Since e−kt = 1 lim a 1 − kt e x →∞ 1 e kt and since the oven is 200oC, we have + 20 = a(1 – 0) + 20 = a + 20 = 200 So a = 180. Thus the formula is T = 180(1 – e−kt) + 20 b) If the temperature of the yam is initially increasing at 2oC per minute, find k. Solution: What this means is that T '(0) = 2. So we need to find T '(t). T(t) = 180 – 180e−kt + 20 = 200 – 180e−kt T '(t) = –180e−kt(−k) = 180ke−kt T '(0) = 2 = 180ke−k(0) = 180k(1) = 180k So k = 1 90 Thus the formula is T = 180(1 – e − 1 t 90 ) + 20