Answer
(a) The temperature after 45 minutes is $137^{\circ}F$
(b) The turkey will have cooled to $100^{\circ}F$ after 116 minutes.
Work Step by Step
(a) We can find $k$:
$T(t) = 75+110~e^{kt}$
$T(30) = 75+110~e^{30k} = 150$
$110~e^{30k} = 75$
$e^{30k} = \frac{75}{110}$
$30k = ln(\frac{75}{110})$
$k = \frac{ln(\frac{75}{110})}{30}$
$k = -0.0127664$
Then:
$T(t) = 75+110~e^{-0.0127664~t}$
We can find the temperature after 45 minutes:
$T(t) = 75+110~e^{-0.0127664~t}$
$T(45) = 75+110~e^{(-0.0127664)(45)}$
$T(45) = 137$
The temperature after 45 minutes is $137^{\circ}F$
(b) We can find the time $t$ when the temperature is $100^{\circ}F$:
$75+110~e^{-0.0127664~t} = 100$
$e^{-0.0127664~t} = \frac{25}{110}$
$-0.0127664~t = ln(\frac{25}{110})$
$t = \frac{ln(\frac{25}{110})}{-0.0127664}$
$t = 116~minutes$
The turkey will have cooled to $100^{\circ}F$ after 116 minutes.
