Question

In a room where the temperature is ${30}^{o}C$, a body cools from ${61}^{o}C$ to ${59}^{o}C$ in 4 minutes. The time taken by the body to cool from ${51}^{o}C$ to ${49}^{o}C$ will be

• A. 4 min
• B. 5 min
• C. 6 min
• D. 8 min

Answer 1

The correct option is C 6 min

The equation given by Newton's law of cooling:
$\frac{dT}{dt}=-bA(T-T_0)$
$\frac{dT}{T-T_0}=-bAdt$ integrating between appropriate limits.
$\frac{1}{t}ln\left(\frac{T_2-T_0}{T_1-T_0}\right)=constant$
Substituting the appropriate conditions:
$\frac{1}{4}ln\left(\frac{59-30}{61-30}\right)=\frac{1}{t}ln\left(\frac{49-30}{51-30}\right)$
Solving for $t$, we get $t=6$ minutes.