Respuesta :
Answer:
123 minutes.
Step-by-step 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: [tex]T(t)=68e^{-0.0174t}+72[/tex].
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.
[tex]80=68e^{-0.0174t}+72[/tex]
Now let us solve for t.
Let us subtract 72 from both sides of our equation.
[tex]80-72=68e^{-0.0174t}+72-72[/tex]
[tex]8=68e^{-0.0174t}[/tex]
Let us divide both sides of our equation by 68.
[tex]\frac{8}{68}=\frac{68e^{-0.0174t}}{68}[/tex]
[tex]0.1176470588235294=e^{-0.0174t}[/tex]
Now let us take natural log of both sides of our equation.
[tex]ln(0.1176470588235294)=ln(e^{-0.0174t})[/tex]
Using natural log property [tex]ln(e^x)=x*ln(e)[/tex] we will get,
[tex]ln(0.1176470588235294)=-0.0174t*ln(e)[/tex]
Since ln(e)=1 , so we will get,
[tex]-2.1400661634962708708=-0.0174t*1[/tex]
[tex]-2.1400661634962708708=-0.0174t[/tex]
Let us divide both sides of our equation by -0.0174.
[tex]\frac{-2.1400661634962708708}{-0.0174}=\frac{-0.0174t}{-0.0174}[/tex]
[tex]122.9923=t[/tex]
[tex]t\approx 123[/tex]
Therefore, it will take approximately 123 minutes for the pie to cool to 80∘F.
