SOLUTION: I would appreciate any help that can be offered on this problem: EXPONENTIAL REGRESSION Data: A cup of hot coffee was placed in a room maintained at a constant temperature of

Question 1039863: I would appreciate any help that can be offered on this problem:
EXPONENTIAL REGRESSION
Data: A cup of hot coffee was placed in a room maintained at a constant temperature of 72 degrees, and the coffee temperature was recorded periodically, in Table 1.
t = Time Elapsed
(minutes) C = Coffee
Temperature (degrees F.)
0 169.0
10 143.5
20 128.2
30 113.3
40 107.5
50 101.4
60 96.9
TABLE 1 REMARKS:
Common sense tells us that the coffee will be cooling off and its temperature will decrease and approach the ambient temperature of the room, 72 degrees.
So, the temperature difference between the coffee temperature and the room temperature will decrease to 0.
We will fit the temperature difference data (Table 2) to an exponential curve of the form y = A e^-bt.
Notice that as t gets large, y will get closer and closer to 0, which is what the temperature difference will do.
So, we want to analyze the data where t = time elapsed and y = C - 72, the temperature difference between the coffee temperature and the room temperature.
TABLE 2
t = Time Elapsed (minutes) y = C - 72 Temperature
Difference
(degrees F.)
0 97.0
10 71.5
20 56.2
30 41.3
40 35.5
50 29.4
60 24.9

Exponential Function of Best Fit (using the data in Table 2):
y = 89.976 e^-.023t where t = Time Elapsed (minutes) and y = Temperature Difference (in degrees)
(a) Use the exponential function to estimate the temperature difference y when 15 minutes have elapsed. Report your estimated temperature difference to the nearest tenth of a degree.

(b) Since y = C - 72, we have coffee temperature C = y + 72. Take your difference estimate from part (a) and add 72 degrees. Interpret the result by filling in the blank:
When 15 minutes have elapsed, the estimated coffee temperature is ________ degrees.
(c) Suppose the coffee temperature C is 105 degrees. Then y = C - 72 = ____ degrees is the temperature difference between the coffee and room temperatures.
(d) Consider the equation _____ = 89.976 e^-.023t where the ____ is filled in with your answer from part (c).

I really appreciate any help that can be offered on this, thank you so much!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

a. Substitute 15 for t in the exponential equation and then do the indicated arithmetic. Given the precision shown in the table, you can probably round to the nearest 10th of a degree.

John

My calculator said it, I believe it, that settles it

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