Question

After the release of radioactive material into the atmosphere from a

nuclear power plant the hay in that country was contaminated by a

radioactive isotope (half-life 88 days). If it is safe to feed the hay

to cows when 15% of the radioactive isotope remains, how long

did the farmers need to wait to use this hay? Round to the nearest

day.

Answer 1

The farmers had to wait 241 days

Explanation:

The exponential decay model is

y = A e^(-k x)

where

• k is the decaying constant
• x is the time it takes the radioactive material to decay
• y is the amount of radioactive isotope that is present

Step 1:

We first need to determine the decaying constant.

1/2 = (1) e^(-88k)

ln(1/2) = -88k

-88k = -ln(2)

k = ln(2) / 88k

k = 7.88×10⁻³

Therefore, the exponential decay model is

y =A e^(-(7.88×10⁻³) x)

Step 2:

We must calculate the the time, x, by using the given information:

y =A e^(-(7.88×10⁻³) x)

0.15 = (1) e^(-(7.88×10⁻³) x)

ln(0.15) = -(7.88×10⁻³) x

x = ln(0.15) / -(7.88×10⁻³)

x = 241 days

Therefore, the farmers had to wait 241 days in order to use this hay.