Answer:
The number of years is approximately 23 years.
Step-by-step explanation:
Given : A certain isotope decays so that the amount A remaining after t years is given by : [tex]A = A_0\cdot e^{-0.03t}[/tex] where [tex]A_0[/tex] is the original amount of the isotope.
To find : To the nearest year, the half-life of the isotope (the amount of time it takes to decay to half the original amount) is how many years?
Solution :
The decay model is given by [tex]A = A_0\cdot e^{-0.03t}[/tex]
We have given that,
The amount of time it takes to decay to half the original amount.
i.e. [tex]A=\frac{A_0}{2}[/tex]
Substitute the values in the formula,
[tex]A=A_0e^{-0.03t}[/tex]
[tex]\frac{A_0}{2}=A_0e^{-(0.03)t}[/tex]
[tex]\frac{1}{2}=e^{-(0.03)t}[/tex]
Taking natural log both side,
[tex]\ln\frac{1}{2}=\ln e^{-(0.03)t}[/tex]
[tex]-\ln2=-(0.03)t\ln e[/tex]
[tex]t=\frac{-\ln2}{-0.03}[/tex]
[tex]t=23.10[/tex]
Therefore, The number of years is approximately 23 years.
