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A body cools from 80C to 70C in 5 minutes and to 62C in the next 5 minutes. Calculate temperature of the surrounding.

Answer
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Hint:Newton’s law of cooling says that the rate of change of the temperature with respect to time is equal to the difference of the temperature of an object and the temperature of the surrounding. The change in temperature decreases as the difference in temperature of the surrounding and the object decreases.

Formula used:The formula of change in temperature with time is given by,
dTdt=k(TTo)
Where average temperature is T, the time is t, the constant is k and the temperature of the surrounding isTo.

Complete step by step solution:
It is given in the problem that a body cools from 80C to 70C in 5 minutes and to 62C in the next 5 minutes and we need to calculate the temperature of the surrounding.
For first decrease in temperature. The average temperature is equal to,
T=80+702
T=1502
T=75C
The formula of change in temperature with time is given by,
dTdt=k(TTo)
Where average temperature is T, the time is t, the constant is k and the temperature of the surrounding isTo.
The body cools from 80C to 70C in 5 minutes. According to Newton’s law of cooling.
80705=k(75To)
105=k(75To)
2=k(75To)………eq. (1)
For the decrease temperature from 70C to 62C in 5 minutes.
The average temperature is equal to,
T=70+622
T=66C
The body cools from 70C to 62C in 5 minutes. According to Newton’s law of cooling.
dTdt=k(TTo)
70625=k(66To)
85=k(66To)………eq. (2)
Dividing the equation (2) with equation (1) we get,
810=66To75To
8(75To)=10(66To)
6008To=66010To
2To=60
To=30C.

The surrounding temperature is equal toTo=30C.

Note:It is advisable for students to understand and remember the formula of Newton's law of cooling as it is very helpful in solving problems like these. The average temperature is taken of the initial and final temperature of the object.