Answer:
18569.234 years
Step-by-step 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[tex]A(t) = A0e^{-rt}[/tex]
[tex]A_0[/tex] = 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,[tex]A(t)=10\% A_0[/tex]
[tex]A(t)=0.1 A_0[/tex]
r = 0.0124 % = 0.000124
Substitute the values in the equation:
[tex]0.1 A_0 = A0e^{- 0.000124t}[/tex]
[tex]- 0.000124t = ln [ \frac{0.1 A_0}{ A_0}][/tex]
[tex]t = ln [ \frac{0.1 A_0}{ A_0}] \times \frac{1}{- 0.000124}[/tex]
[tex]t = ln [0.1] \times \frac{1}{-0.000124}[/tex]
[tex]t = 18569.234[/tex]
Hence it will take 18569.234 years to decay to 10 percent of its original amount.
