College Algebra & Trigonometry, 2018a

3.5. APPLICATIONS OF THE NEGATIVE EXPONENTIAL FUNCTION 183
T = T a + Ce −kt
72 = 44 + Ce −k∗0
72 = 44 + Ce 0 =44+C ∗ 1
72 = 44 + C
28 = C
Now we know that T a =44and C =28. We can use the other piece of information
from the problem to calculate the value of k. The problem states that after 30
minutes the soda has cooled off to 61 ◦ F. That means that when t =30(or t =0.5
depending on which units you choose) the temperature T willbe61 ◦ F. We can
then set up the equation to reflect this and calculate the vlaue of k:
T =44+28e −kt
61 = 44 + 28e −k∗30
First, we’ll subtract 44 on both sides and divide by 28:
61 = 44 + 28e −k∗30
17 = 28e −30k
17
28 = e−30k