Question

A hot body, obeying Newton's law of cooling is cooling down from its peak value 80${}^{\circ }$C to an ambient temperature of 30${}^{\circ }$C. It takes 5 minutes in cooling down from 80${}^{\circ }$C to 40${}^{\circ }$C. How much time will it take to cool down from 62${}^{\circ }$C to 32${}^{\circ }$C ?
(Given In 2 = 0.693, In 5 = 1.609)

• A. 3.75 minutes
• B. 8.6 minutes
• C. 9.6 minutes
• D. 6.5 minutes

Answer 1

Answer: (B)

$\frac{d\theta }{dt}=-C\left(\theta -{\theta }_0\right)$
$\underset{80}{\overset{40}{\int }}\frac{1}{\theta -{\theta }_0}d\theta =-C/5$
$\underset{62}{\overset{32}{\int }}\frac{1}{\theta -{\theta }_0}d\theta =-Ct$
$ln\left(\frac{80-30}{40-30}\right)=5C$
$In\left(\frac{62-30}{40-50}\right)=Ct$
$In5=5c=1.609$
$In16=ct=4\times 0.693$
$t=8.6min$