Question

Hot custard in heated at a temperature of 194°F. It is then allowed to cool in the refrigerator at a temperature of 41°F. The custard cools to 140°F in 20 minutes. What will be the temperature of the custard after 80 minutes?

guys pls help

Answer 1

To solve this problem, we can use Newton's Law of Cooling, which states that the rate of cooling of an object is proportional to the difference in temperature between the object and its surroundings.

Let T(t) be the temperature of the custard at time t. Then we have:

T'(t) = k(T(t) - 41)

where k is a constant of proportionality. We know that T(0) = 194 and T(20) = 140, so we can solve for k:

k = (T(0) - 41) / (T(0) - T(20)) = (194 - 41) / (194 - 140) = 3.25

Now we can use this value of k to find T(80):

T'(t) = 3.25(T(t) - 41)
T'(t) / (T(t) - 41) = 3.25
ln|T(t) - 41| = 3.25t + C
T(t) - 41 = e^(3.25t + C)
T(t) = e^(3.25t + C) + 41

Using the initial condition T(0) = 194, we can solve for C:

194 = e^C + 41
C = ln(153)

So the temperature of the custard after 80 minutes is:

T(80) = e^(3.25*80 + ln(153)) + 41 = 51.33°F

Therefore, the temperature of the custard after 80 minutes is 51.33°F.