Answer :
Final answer:
Assuming a linear cooling rate and using Newton's Law of Cooling for approximation, it will take approximately 47 minutes and 10 seconds for the cake to cool to 100 degrees Fahrenheit.
Explanation:
The cooling of the cake can be modeled as an exponential decay problem.
Using Newton's Law of Cooling, we might say that the cake cools down at a rate proportional to the difference between its temperature and that of the room.
Let's say that it takes t minutes for the cake to reach 100 degrees F.
If we suppose that 30 minutes into cooling the cake's temperature was 140 degrees F, we can use this as a known data point to help construct our exponential decay model.
To calculate this precisely we would need the exact cooling rate, but assuming a linear decay (which isn't strictly true but can serve as a close approximation here),
we can calculate it as follows: the drop from 210 to 140 degrees is 70 degrees in 30 minutes.
That suggests a cooling rate of 70/30 = 2.33 degrees per minute.
If we use that rate, when the cake reaches 100 degrees it will have cooled a further (140-100) = 40 degrees.
At a rate of 2.33 degrees per minute, it'll take roughly a further 40/2.33 = 17.167, or about 17 minutes 10 seconds.
So,using the linear model under this assumption, it will take roughly 30 (initial cooling period) + 17 (to reach 100 degrees) = 47 minutes 10 seconds total.
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