College Algebra & Trigonometry, 2018a

3.5. APPLICATIONS OF THE NEGATIVE EXPONENTIAL FUNCTION 181
Now that we know the value of k, we can directly calculate the amount of 198 Au
left after 6 days:
A(t) =A 0 e −kt
=65e −0.2567∗6
≈ 13.932
So, approximately 13.932 grams of 198 Au would be left after 6 days.
In order to calculate how long it takes for 10 grams of 198 Au to be left, we’ll need
to solve for t in the equation A(t) =A 0 e −kt with A(t) =10:
A(t) =A 0 e −kt
10 = 65e −0.2567t
First, we’ll divide on both sides by 65:
10
65 = ✚✚ 65e −0.2567t
✚ ✚ 65
10
65 = e−0.2567t
Then, take the natural logarithm on both sides:
ln
( ) 10
=ln ( e −0.2567t)
65