Question

The temperature T of a cup of coffee is a function T(t) where t is the time in minutes. The room temperature is 18° Celsius. The rate at which the coffee cools down is proportional to the difference between the temperature of the coffee and the room temperature. Use this information to write a differential equation describing the derivative of the coffee temperature in terms of T and t. Use C as your proportionality constant. C should be a positive number. Write T instead of T(t).

Answer 1

Answer:

(1/C)dR/dt - dT/dt = 0

Step-by-step explanation:

Let R be the rate at which the coffee cools down. Since R is proportional to the difference between the temperature of the coffee, T and the room temperature, we write

R = C(T - 18)

Where C is the proportionality constant.

To write a differential equation describing the derivative of the coffee temperature in terms of T and t, we need to differentiate

R = C(T - 18)

with respect to t.

Doing that, we have

dR/dt = CdT/dt

Or

(1/C)dR/dt - dT/dt = 0

Which is the required differential equation.