At time t = 0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius (°C) at time t = 0, and the internal temperature of the potato is greater than 27°C for all times t > 0. The internal temperature of the potato at time t minutes can be modeled by the function H that satisfies the differential equation dH/dT=−1/4(H−27), where H() is measured in degrees Celsius and H(0) = 91.
(a) Write an equation for the line tangent to the graph of H at t = 0. Use this equation to approximate the internal temperature of the potato at time t = 3
(b)Use d2H/dT2 to determine whether your answer in part (a) is an underestimate or a overestimate of the internal temperature of the potato at time t = 3.
(c) For t < 10, an alternate model for the internal temperature of the potato at time t minutes is the function G that satisfies the differential equation G/dT=−(−27)2/3, where G(t) is measured in degrees Celsius and G(0) = 91. Find an expression for G(t). Based on this model, what is the internal temperature of the potato at time t = 3?