Question

A body cools from $75{}^\circ C$ to ${{65}^{\circ }}C$ in 5 minutes. If the room temperature is $25{}^\circ C$ then the temperature of the body at the end of next 5 minutes is:
1. $57{}^\circ C$
2. $55{}^\circ C$
3. $54{}^\circ C$
4. $53{}^\circ C$

Answer 1

Hint: We will use the concept of Newton's law of cooling and its mathematical representation in order to solve this equation. Here we have to find the temperature of the body at the end of the next five minutes.

Formula used:
$t=\frac{1}{k}\ln \left( \frac{{{\theta }_{2}}-{{\theta }_{n}}}{{{\theta }_{1}}-{{\theta }_{n}}} \right)$

Complete answer:
Newton’s law of cooling states that the rate of loss of heat from a body is directly proportional to the temperature difference between the body and its surroundings. That is, hot tea cools faster than warm tea. Newton’s law of cooling actually explains the rate of cooling.

From Newton’s law of cooling, we have the equation as
Time, $t=\frac{1}{k}\ln \left( \frac{{{\theta }_{2}}-{{\theta }_{n}}}{{{\theta }_{1}}-{{\theta }_{n}}} \right)$

Given in the question that a body cools from $75{}^\circ C$ to $65{}^\circ C$ in 5 minutes and the room temperature is $25{}^\circ C$
$5=\frac{1}{k}\ln \left( \frac{65-25}{75-25} \right)-(1)$

Let $\theta$ be the final temperature after the next five minutes. Then we have the equation as:
$5=\frac{1}{k}\ln \left( \frac{\theta -25}{65-25} \right)-(2)$
On dividing equation (2) by equation (1), we get:
$1=\frac{\ln \left( \frac{\theta -25}{40} \right)}{\ln \left( \frac{40}{50} \right)}$

Taking antilog and then solving the equation we get the temperature of the body in the next five minutes as:
Temperature, $\theta =57{}^\circ C$

Therefore, the answer is option (1)

Note: For the second case you should remember that the initial temperature in this case is $65{}^\circ C$as it is the temperature the body reaches after the first five minutes. K in the equation is just a positive constant which is the same in both the cases since the body under observation is the same for both cases. K depends on the area and nature of the surface of the body. This problem can also be solved in different ways.