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 : \(A = A_0\cdot e^{-0.03t}\) where \(A_0\) 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 \(A = A_0\cdot e^{-0.03t}\)
We have given that,
The amount of time it takes to decay to half the original amount.
i.e. \(A=\frac{A_0}{2}\)
Substitute the values in the formula,
\(A=A_0e^{-0.03t}\)
\(\frac{A_0}{2}=A_0e^{-(0.03)t}\)
\(\frac{1}{2}=e^{-(0.03)t}\)
Taking natural log both side,
\(\ln\frac{1}{2}=\ln e^{-(0.03)t}\)
\(-\ln2=-(0.03)t\ln e\)
\(t=\frac{-\ln2}{-0.03}\)
\(t=23.10\)
Therefore, The number of years is approximately 23 years.
