Solve the problem.
-A company wishes to analyze a new room deodorizer. The active ingredient evaporates at a rate proportional to the amount present. One third of the ingredient evaporates in the first 20 days after the deodorizer is installed. If
The deodorizer becomes ineffective after 85% of the active ingredient has evaporated, how long will one of these
Deodorizers remain effective? Round to the nearest whole number as needed.
A) 26 days
B) 45 days
C) 94 days
D) 100 days

Question

Quizplus · Accepted Answer

Let's use the formula for exponential decay:
A = A0e^-kt, where:
A = amount remaining after time t
A0 = initial amount
k = decay constant
t = time

We know that after 20 days, one third of the active ingredient has evaporated, so we can set up the following equation:
2/3 A0 = A0e^-20k

Simplifying this, we get:
e^20k = 3/2
20k = ln(3/2)
k = (1/20)ln(3/2)

Now we can use the fact that the deodorizer becomes ineffective after 85% of the active ingredient has evaporated:
0.15 A0 = A0e^-kt

Simplifying this, we get:
e^kt = 20/3
t = (1/k) ln(20/3)

Plugging in our value for k, we get:
t = (20/ln(3/2)) ln(20/3)
t ≈ 94 days

Therefore, the best choice is C, 94 days.

Solve the Problem. -A Company Wishes to Analyze a New Room Deodorizer. the Room