Newton's law of cooling

Question

How many minutes elapse before an object with a temperature 525 degrees fahrenheit reaches 100 degrees fahrenheit in a room that is 85 degree fahren heit. Assume the cooling constant is 0.031.

Answer 1

Newton's Law of Cooling goes like this:

T(t) = Ce^kt + A where T(t) is the temperature of the item cooling at any time t, C is a constant representing the difference in the initial temperature and the ambient temperature, and k is the cooling constant (a negative value).

In this case the C = 525 - 85 = 440, A = 85, and we need to add a negative in front of the given cooling constant so it becomes -0.031 making the function:

T(t) = 440e^-0.031t + 85

We are trying to determine t when T = 100 so:

100 = 440e^-0.031t + 85 [subtract 85 from both sides to get:

15 = 440e^-0.031t [divide both sides by 440 to get:

(3/88) = e^-0.031t [take the natural log of both sides to get:

ln(3/88) = ln(e^-0.031t) [use the property of logarithms that allows you to move the exponent in front of the log function as a multiplier to get:

ln(3/88) = (-0.031t)•ln(e) [[ln(e) = 1 so:

ln(3/88) = (-0.031t) [divide both sides by -0.031 to get:

t = 108.99 or, 109 minutes