College Algebra & Trigonometry, 2018a

182 CHAPTER 3. EXPONENTS AND LOGARITHMS
We’ll calculate an approximate value for ln ( 10
65)
and restate the right hand side of
the equation using the properties of logarithms:
−1.872 ≈−0.2567t ∗ ln e
−1.872 ≈−0.2567t ∗ 1
−1.872 ≈−0.2567t
7.3 ≈ t
So, it would take about 7.3 days for the 65 grams of 198 Au to decay to 10 grams.
Newton’s Law of Cooling
Newton’s Law of Cooling states that the temperature of an object can be determined
using the equation:
T = T a + Ce −kt
where T a is the ambient temperature of the surrounding environment. The values
of the constants C and k can often be calculated from given information.
Example
A bottle of soda at room temperature (72 ◦ F) is placed in a refrigerator where the
temperature is 44 ◦ F.
After half an hour, the soda has cooled to 61 ◦ F. What is the temperature of the
soda after another half hour?
First, since the soda is placed in the refrigerator where the ambient temperature is
44 ◦ F, then T a =44. We also know that at t =0, T =72, which is the temperature of
the soda can when it is first put in the refrigerator. This will allow us to calculate
the constant C.