Newton's law of cooling states that the rate of change in the temperature of an object is proportional to the difference between the temperature of the object and the ambient temperature.

Let y be the temperature of the turkey. Then `y'=k(y-75)`

`(dy)/(dt)=k(y-75)`

`(dy)/(y-75)=kdt`  Integrating we get:

`ln|y-75|=kt+C_1` Since y>75 we can lose the absolute value signs. Now exponentiate each side:

`y-75=e^(kt+C_1)=e^(kt)e^(C_1)=Ce^(kt)`

So `y=Ce^(kt)+75`

We are given two data points: (0,185) and (30,150) where x is in minutes and y is in degrees F.

Using (0,185) we see that `185=C+75==>C=110`

Using (30,150) and C=110 we get `150=110e^(30k)+75`

`75=110e^(30k)`

`e^(30k)=15/22`

`30k=ln(15/22)`

`k=(ln(15/22))/30~~.-.013`

So `y=110e^(-.013t)+75`

(a) If t=40 then `y=110e^(-.52)+75~~141`

So the temperature is approximately `141^@`

(b) If y=110 then `110=110e^(-.013t)+75`

`35=110e^(-.013t)`

`e^(-.013t)=7/22`

`-.013t=ln(7/22)`

`t=(ln(7/22))/-.013~~88` minutes

So it will take approximately 88 minutes.