Answer
(a) The coffee cools most quickly at the start when the temperature difference is at a maximum. As time goes by, the rate of cooling decreases as the coffee's temperature gets closer to the surrounding temperature.
(b) Differential equation: $T' = k(T-T_s)$
$T'$ is the rate of temperature change
$k$ is the relative rate of temperature change, which is a constant
$T$ is the temperature
$T_s$ is the surrounding temperature
(c) Please see the graph below.
Work Step by Step
(a) The coffee cools most quickly at the start when the temperature difference is at a maximum. As time goes by, the rate of cooling decreases as the coffee's temperature gets closer to the surrounding temperature.
(b) Differential equation: $T' = k(T-T_s)$
$T'$ is the rate of temperature change
$k$ is the relative rate of temperature change, which is a constant
$T$ is the temperature
$T_s$ is the surrounding temperature
This differential equation seems appropriate since the rate of cooling decreases as the value of $T-T_s$ decreases. That is, the rate of cooling decreases as the coffee's temperature gets closer to the surrounding temperature.
(c) Please see the graph below.
