Chapter 6.2, Problem 66E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### 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 20Â°F and the core temperature is 160Â°F and the temperature of the liquid is 60Â°F 5 minutes after it is removed.

Formula used:

ifln(x)=ythenx=ey

Integration of xn given as,

âˆ«xndx=xn+1n+1+C

Integration of 1/x given by,

âˆ«1xdx=log|x|+C

Calculation:

According to the Newtonâ€™s law of cooling,

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

dydt=k(yâˆ’20)

Next by integrating the above equation:

âˆ«dydt=âˆ«k(yâˆ’20)

Hence,

âˆ«dyyâˆ’20=âˆ«kdt

Integrating the above equation,

ln|yâˆ’20|=kt+

(b)

To determine

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

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