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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Newton's Law of Cooling A container of hot liquid is placed in a freezer that is kept at a constant temperature of 20 F . The initial temperature of the liquid is 160 F . After 5 minutes, the liquid’s temperature is 60 F .

(a) Write an equation for the temperature y of the liquid t minutes after it is placed in the freezer.

(b) How much longer will it take for the temperature of the liquid to decrease to 25 F?

(a)

To determine

To calculate: An equation for the temperature y of a liquid after it is placed in the freezer for t minutes.

Explanation

Given:

A container of hot liquid is kept in a freezer at a constant temperature of 20°F and the core temperature is 160°F and the temperature of the liquid is 60°F 5 minutes after it is removed.

Formula used:

ifln(x)=ythenx=ey

Integration of xn given as,

xndx=xn+1n+1+C

Integration of 1/x given by,

1xdx=log|x|+C

Calculation:

According to the Newton’s law of cooling,

The rate of temperature will be proportional to the difference between y and constant temperature 20°F.

dydt=k(y20)

Next by integrating the above equation:

dydt=k(y20)

Hence,

dyy20=kdt

Integrating the above equation,

ln|y20|=kt+

(b)

To determine

To calculate: The time taken for the temperature of liquid to decrease to 25°F.

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