A cake recipe says to bake a cake until the center is 180 degrees then let it cool to 110 degrees. If I have a table that says at 5 minutes the cake is 124 degrees, at 10 minutes the cake is 92 degrees, at 15 minutes the cake is 78 degrees and at 20 minutes the cake is 70 degrees. What is an exponential model for this data if the room temperature is 68 degrees
To find an exponential model for the given data, we can use the general form of an exponential function:
y = A * e^(kt)
where:
y is the temperature of the cake at a given time t,
A is the initial temperature of the cake,
e is Euler's number (~2.71828),
k is the rate of cooling, and
t is the time in minutes.
From the given data, we can obtain the following equations:
124 = A * e^(k * 5)
92 = A * e^(k * 10)
78 = A * e^(k * 15)
70 = A * e^(k * 20)
We can divide the equations to eliminate A:
124/92 = e^(k * 5) / e^(k * 10) => (124/92) = e^(k * (5-10)) => e^(-5k) = 124/92
92/78 = e^(k * 10) / e^(k * 15) => (92/78) = e^(k * (10-15)) => e^(-5k) = 92/78
78/70 = e^(k * 15) / e^(k * 20) => (78/70) = e^(k * (15-20)) => e^(-5k) = 78/70
Now, we can equate the expressions for e^(-5k):
124/92 = 92/78 => 124 * 78 = 92 * 92 => 9672 = 8464
Clearly, this is not true. Therefore, there is no exponential model that exactly fits the given data.