Question

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(t) is measured in degrees Celsius and H(0)=91.

(b) Use (d²H/dt²) to determine whether your answer in Part A is an underestimate or an overestimate of the internal temperature of the potato at time t=3.

Answer 1

The answer in Part A is an overestimate of the internal temperature of the potato at time t=3.

Step-by-step explanation:

The differential equation given in the question is (dH/dt) = -(1/4)(H - 27), where H(t) is the internal temperature of the potato at time t. This equation models the rate at which the internal temperature of the potato changes over time. To determine whether the given answer in Part A is an underestimate or an overestimate of the internal temperature of the potato at time t=3, we can use the second derivative of H with respect to t (d²H/dt²).

By taking the second derivative of H, we get d²H/dt² = (1/4)(dH/dt). Since the given differential equation (dH/dt) = -(1/4)(H - 27) is linear and does not involve higher derivatives, the second derivative is equal to a constant multiple of the first derivative. This means that the second derivative has the same sign and direction as the first derivative.

Since the first derivative is negative for all values of H and t in the given problem, this implies that the second derivative is also negative. Therefore, the internal temperature of the potato at time t=3 is decreasing, and the answer in Part A is an overestimate of the internal temperature.