A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 53°F

(a) If the temperature of the turkey is 150°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)

(b) When will the turkey cool to 100°F? (Round your answer to the nearest whole number.)

The temperature will drop on an exponential decay curve, i.e. a curve with the equation

F = 53 + (185-53)exp(-a.t) = 53 + 132.exp(-a.t)
for some parameter a which we need to determine. We know that because at t = 0, F = 185, and as t increases without limit, F tends to 53. We know from (a) that 150 = 53 + 132.exp(-30a), so -30a = ln((150-53)/(185-53)) = -0.05239, so a = 0.01027. So the temperature after 45 mins should be:
53 + 132.exp(-0.01027 x 45) = 136 degrees to the nearest degree.
The turkey cools to 100 degrees F when 100 = 53 + 132.exp(-0.01027 x t)
and that will be when ln((100-53)/132) = -0.01027 x t,
i.e. t = -ln((100-53)/132) / 0.01027 = 101 minutes to the nearest minute.

To answer these questions, we can use the concept of heat transfer and the principle of exponential decay.

(a) To find the turkey's temperature after 45 minutes, we need to understand how the turkey cools down over time. The rate at which it cools is proportional to the temperature difference between the turkey and its surroundings. We can use the formula:

ΔT = (T_initial - T_final) * e^(-kt)

where:
- ΔT is the change in temperature over time,
- T_initial is the initial temperature of the turkey,
- T_final is the temperature of the environment (room temperature),
- k is a proportionality constant, and
- t is the time.

Since we know the initial temperature (150°F), the final temperature of the environment (53°F), and the time (45 minutes), we can rearrange the formula to solve for the turkey's temperature after 45 minutes.

Let's plug in the known values:

150 - 53 = (150 - T_final) * e^(-k * 45)

Simplifying, we get:

97 = (150 - T_final) * e^(-45k)

We can solve this equation to find k:

97 / (150 - T_final) = e^(-45k)

Taking the natural logarithm of both sides, we have:

ln(97 / (150 - T_final)) = -45k

Now we can solve for k:

k = ln(97 / (150 - T_final)) / -45

By substituting this value of k back into our equation, we can find the turkey's temperature after 45 minutes.

(b) To determine when the turkey will cool to 100°F, we can apply a similar approach:

100 = (150 - 53) * e^(-kt)

Let's plug in the values:

47 = 97 * e^(-kt)

Rearranging the equation, we get:

e^(-kt) = 47 / 97

Again, let's take the natural logarithm of both sides:

-kt = ln(47 / 97)

Solving for t, we get:

t = ln(47 / 97) / -k

Substituting the previously calculated value of k, we can find the time it will take for the turkey to cool to 100°F.

Note: The values of k and t will depend on the specific heat transfer properties of the turkey and the environment.