Question

A hot metal sphere cools from ${60}^{0}C$ to ${52}^{0}C$ in 5 minutes and from ${52}^{0}C$ to ${44}^{0}C$ in next 7.5 minutes.Determine ts temperature in the next 10 minutes.

• A. ${58}^{0}C$
• B. ${78}^{0}C$
• C. ${38}^{0}C$
• D. $8^{0}C$

Answer 1

The correct option is C ${38}^{0}C$

We know that,

$\frac{d\theta }{dt}\propto ({\theta }_t-{\theta }_s)$

$\frac{d\theta }{dt}=k({\theta }_t-{\theta }_s)$ --------------------------(1)

$dt=5min$

$d\theta =60-52=8^{\circ }C$

${\theta }_t=\frac{60+52}{2}={56}^{\circ }C$

Substitute above value in equation (1) we will get

$\frac{8}{5}=k(56-{\theta }_s)$-----------------------------------(2)

Similarly,

$dt=7.5min$

$d\theta =52-44=8^{\circ }C$

${\theta }_t=\frac{52+44}{2}={48}^{\circ }C$

Substitute above value in equation (1) we will get

$\frac{8}{7.5}=k(48-{\theta }_s)$-----------------------------------(3)

By Dividing both the above equation we will get

$\frac{7.5}{5}=\frac{k(56-{\theta }_s)}{k(48-{\theta }_s)}$

$7.5(48-{\theta }_s)=5(56-{\theta }_s)$

$360-{7.5\theta }_s=280-{5\theta }_s$

$360-280=7.5{\theta }_s-{5\theta }_s$

${2.5\theta }_s=80$

${\theta }_s=32$

Substitute above value in equation (2)

$k=\frac{1}{15}$

$dt=10min$

$d\theta =44-T$

${\theta }_t=\frac{44+T}{2}$

Substitute all value in equation (1)we will get

$T={38}^{\circ }C$

Hence, option (C) is correct option