Question

Carbon–14 is a radioactive isotope that decays exponentially at a rate of 0.0124 percent a year. How many years will it take for carbon–14 to decay to 10 percent of its original amount? The equation for exponential decay is At = A0e–rt.

Answer 1

18569.234 years

Explanation:

Given : Carbon–14 is a radioactive isotope that decays exponentially at a rate of 0.0124 percent a year.

To Find: How many years will it take for carbon–14 to decay to 10 percent of its original amount?

Solution:

The equation for exponential decay is
`A(t) = A0e^(-rt)`

`A_0` = initial amount

A(t) = Amount after t time

Now we are supposed to find after how many years will it take for carbon–14 to decay to 10 percent of its original amount.

So,
`A(t)=10\% A_0`

`A(t)=0.1 A_0`

r = 0.0124 % = 0.000124

Substitute the values in the equation:

`0.1 A_0 = A0e^(- 0.000124t)`

`- 0.000124t = ln [ (0.1 A_0)/( A_0)]`

`t = ln [ (0.1 A_0)/( A_0)] * (1)/(- 0.000124)`

`t = ln [0.1] * (1)/(-0.000124)`

`t = 18569.234`

Hence it will take 18569.234 years to decay to 10 percent of its original amount.