Final answer:
Using Newton's Law of Cooling, we find the cooling constant k with given temperatures and solve for the time it takes for the cake to cool to 100° F.
Step-by-step explanation:
To solve when the cake will reach 100° F after being removed from the oven at 210° F and cooling in a room at 70° F, we'll need to use the concept of exponential decay, often used in physics and engineering to model cooling processes. This situation can be described with Newton's Law of Cooling, which tells us that the rate of cooling is proportional to the temperature difference between the object and its surroundings. Mathematically, the temperature of the cake T at time t can be modeled using the equation T(t) = Ts + (T0 - Ts) · e-kt, where Ts is the surrounding temperature, T0 is the initial temperature of the cake, k is the cooling constant, and t is the time.
First, we'll find the cooling constant k using the information that after 30 minutes, the cake's temperature is 140° F. We have the following data: Ts = 70° F, T0 = 210° F, T(30) = 140° F, and t = 30 minutes.
Inserting these into the equation and solving for k gives us:
140 = 70 + (210 - 70) · e-30k
70 = 140 · e-30k
0.5 = e-30k
-30k = ln(0.5)
k = - ln(0.5) / 30
After finding k, we use it to find the time it will take for the cake to reach 100° F:
100 = 70 + (210 - 70) · e-kt
30 = 140 · e-kt
e-kt = 30 / 140
-kt = ln(30 / 140)
t = ln(30 / 140) / k
By calculating t using the cooling constant k, we find the time at which the cake will be 100° F.