Question

A pie with a temperature of 140∘F is taken out of the oven and placed on a windowsill to cool. Its temperature as a function of tminutes is given by

T(t)=68e−0.0174t+72

How long, to the closest minute, will it take for the pie to cool to
80∘F?

Answer 1

123 minutes.

Explanation:

We have been given that a pie with a temperature of 140∘F is taken out of the oven and placed on a windowsill to cool. Its temperature as a function of t minutes is given by:
`T(t)=68e^(-0.0174t)+72`.

To find the number of minutes it will take for the pie to cool to 80∘F, we will substitute T(t) = 80 in our given function.

`80=68e^(-0.0174t)+72`

Now let us solve for t.

Let us subtract 72 from both sides of our equation.

`80-72=68e^(-0.0174t)+72-72`

`8=68e^(-0.0174t)`

Let us divide both sides of our equation by 68.

`(8)/(68)=(68e^(-0.0174t))/(68)`

`0.1176470588235294=e^(-0.0174t)`

Now let us take natural log of both sides of our equation.

`ln(0.1176470588235294)=ln(e^(-0.0174t))`

Using natural log property
`ln(e^x)=x*ln(e)` we will get,

`ln(0.1176470588235294)=-0.0174t*ln(e)`

Since ln(e)=1 , so we will get,

`-2.1400661634962708708=-0.0174t*1`

`-2.1400661634962708708=-0.0174t`

Let us divide both sides of our equation by -0.0174.

`(-2.1400661634962708708)/(-0.0174)=(-0.0174t)/(-0.0174)`

`122.9923=t`

`t\approx 123`

Therefore, it will take approximately 123 minutes for the pie to cool to 80∘F.