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Answer:
47.9758 minutes
Step-by-step explanation:
The temperature difference between the turkey and the room starts at ...
169 °F - 73 °F = 96 °F
and drops to ...
140 °F -73 °F = 67 °F
after 11 minutes. In that time period, the difference becomes 67/96 of what it was initially.
We are asked to find the time it takes for the temperature difference to get to ...
93 °F -73 °F = 20 °F
Then the exponential decay equation we want to solve is ...
20 = 96(67/96)^(t/11) . . . . . t is time in minutes
Dividing by 96 and taking logs, we get ...
20/96 = (67/96)^(t/11)
log(20/96) = (t/11)log(67/96)
Solving for t gives ...
t = 11·log(20/96)/log(67/96) ≈ 47.9758 . . . minutes
After being removed from the oven, it will take the turkey 47.9758 minutes to cool to 93 °F.
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<em>Comment on this method of solution</em>
We have dealt exclusively with the temperature difference, and have written the exponential equation in terms of the fractional change in a given time. The form of the equation we wrote is ...
f(t) = (initial value)×(multiplier)^(t/(multiplier's period))
The "multiplier" may also be called a "decay constant" or "growth constant". The exponential term can also be expressed as a power of e, the base of natural logarithms. You will often see that as ...
e^(-kt)
where k is the natural logarithm:
k = -ln(multiplier)/(multiplier's period)
In this problem, the value of k would be ...
k = -ln(67/96)/11 ≈ 0.0326959611
Using this, the temperature decay would be written as ...
temp difference = 96·e^(-0.0326959611t)
and the solution for a temperature difference of 20 degrees would be ...
ln(20/96)/(-0.0326959611) ≈ 47.9758 . . . minutes
You need at least 6 significant figures in the value of k because the answer has 6 significant figures. (Why, we don't know, as neither temperature nor time measurements can be made to that accuracy in this situation. We think it is so the grader can tell if you worked the problem correctly.)
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You can also write the equation for the turkey temperature (T) as ...
T(t) = 73 +96e^(-0.0326959611t)
This is the room temperature added to the difference from room temperature.
Then you would be seeking the solution to T(t) = 93.
