A certain radioactive substance is decaying at a rate proportional to the amount present. If 100 grams decays to 13.5 grams in 4 years, how long will it take for 90 grams to decay to 30 grams?
A) 2.405 yr
B) 2.195 yr
C) 0.501 yr
D) Problem cannot be solved as stated.

Question

Quizplus · Accepted Answer

Let the amount present at time t be given by A(t). Then, we have:

A(0) = 100 (initial amount)
A(4) = 13.5 (amount after 4 years)

Since the decay is proportional to the amount present, we have:

A'(t) = -kA(t)

where k is the proportionality constant. To solve for k, we use the fact that A(4) = 13.5:

A'(4) = -kA(4)
(13.5 - 100)/4 = -k(13.5)
k = (100 - 13.5)/4(13.5) = 0.0139

Now, we want to find the time it takes for A(t) to decay from 90 to 30 grams:

A(t) = 90
A(0) = 100

We can solve for t by integrating:

ln|A(t)/A(0)| = -kt

ln|90/100| = -0.0139t

t = ln(10/9)/0.0139 ≈ 2.195 years

Therefore, the answer is B, 2.195 yr.

A Certain Radioactive Substance Is Decaying at a Rate Proportional