A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a table in a room where the temperature is 75 Fahrenheit.

Question

Give answers accurate to at least 2 decimal places.

If the temperature of the turkey is 153 Fahrenheit after half an hour, what is its temperature after 45 minutes?

  ( )  Fahrenheit

When will the turkey cool to 100 Fahrenheit?

( ) hours.

Answer 1

Question: 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 75 F. Give answers accurate to at least 2 decimal places.

a. If the temperature of the turkey is 153 Fahrenheit after half an hour, what is its temperature after 45 minutes?

b. When will the turkey cool to 100 Fahrenheit?

Answer: Let's use Newton's Law of Cooling.

We have an ambient temperature T_a of 75 F, and the turkey's initial temperature T₀ is 185 F. We also know that after a half hour, the turkey's temperature is T(0.5) = 153 F.

Here is Newton's Law of Cooling: T(t) - T_a = (T₀ - T_a)e^-kt. T(t) refers to the turkey's temperature at time t.

a. We will need to find a rate k that is dictating the turkey's cooling. But first, Let's plug in what we know:

T(0.5) - (75) = [(185) - (75)]e^-(0.5)k

153 - 75 = (185 - 75)e^-(0.5)k

78 = 110e^-0.5k. We'll need to solve for k, so let's knock off -0.75k from its pedestal. Divide both sides by 110 and use natural logarithms on both sides:

(39/55) = e^-0.5k

ln(39/55) = ln(e^-0.5k)

ln(39/55) = -0.5k. Now divide both sides by -0.≈5.

k ≈ 0.6875431 (I know that's a lot, but it's just to keep accuracy until we get to the final answer.)

^^^Let me recap what I just did: I don't have complete information from what I had in part a of the question. I know that at 30 minutes, the turkey's temperature is 153 F, but I can't exactly guess what it is at 45 minutes. What I do know, however, is that some unknown rate k is driving this cooling process. I need to find that k first before I can apply it to the 45 minute question, or any other question of time for that matter. We just found that k, so let's substitute it into a model for the 45 minute question (remember that 45 minutes is (3/4) hr, or 0.75 hr):

T(t) - T_a = (T₀ - T_a)e^-kt.

T(0.75) - (75) = [(185) - (75)]e^{-0.6875431(0.75)}. We need to solve for T(0.75). Add 75 to both sides.

T(0.75) = 75 + 110e^-0.5156573

T(0.75) = 75 + 110(0.59710799)

T(0.75) = 75 + 65.68187852

T(0.75) = 140.6818785

After 45 minutes, the turkey's temperature is 140.68 F.

b. The heavy lifting of creating a general model in part a allows us to use it for this part.

T(t) - T_a = (T₀ - T_a)e^-kt

Whenever the turkey's temperature cools to 100 F (and who knows then that'll happen, huh?), we can say that T(t) = 100.

(100) - (75) = [(185) - (75)]e^-0.6875431t

100 - 75 = 110e^-0.6875431t

25 = 110e^-0.6875431t. Divide both sides by 110, and then take the natural logarithm of both sides.

(5/22) = e^-0.6875431t

ln(5/22) = ln(e^-0.6875431t)

ln(5/22) = -0.6875431t. Divide both sides by -0.6875431.

t ≈ 2.15492606

The turkey's temperature will be 100 F after 2.15 hours (2 hours, 9 minutes).