Chapter 24, Problem 40P

The rate of cooling of a body (Fig. P24.40) can be expressed as

$\frac{dT}{dt}=-k\left(T-T_a\right)$

Where $$ $T=\text{temperature of the body }\left(°C\right),T_a=\text{temperature of }$ the surrounding medium $\left(°\text{C}\right)$, and $k=a$ a proportionality constant (per minute). Thus, this equation (called Newton's law of cooling) specifies that the rate of cooling is proportional to the difference in the temperatures of the body and of the surrounding medium. If a metal ball heated to $80\text{°C}$ is dropped into water that is held constant at $T_a=20\text{°C}$, the temperature of the ball changes, as in

Time, min	0	5	10	15	20	25
$T,°C$	80	44.5	30.0	24.1	21.7	20.7

Utilize numerical differentiation to determine $dT/dt$ at each value of time. Plot $dT/dt$ versus $T-T_a$ and employ linear regression to evaluate k.

FIGURE P24.40