Question

A radioactive substance decays exponentially. A scientist begins with 130 milligrams of a radioactive substance. After 30 hours, 65 mg of the substance remains. How many milligrams will remain after 56 hours?

Answer 1

f(x) = abˣ

"A scientist begins with 130 milligrams", or we can word it differently.

when x = 0, f(x) = 130, which in short simply means a = 130.

"After 30 hours, 65 mg of the substance remains", worded differently.

when x = 30, f(x) = 65.

`f(x)=ab^x~\hspace{10em}\underline{f(x)=130b^x} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} ~\hfill x&=30\\ f(x)&=65 \end{cases}\qquad \implies 65=130b^(30)\implies \cfrac{65}{130}=b^(30)\implies \cfrac{1}{2}=b^(30)`

`\sqrt[30]{\cfrac{1}{2}}=b\implies \cfrac{\sqrt[30]{1}}{\sqrt[30]{2}}=b\implies \cfrac{1}{\sqrt[30]{2}}=b~\hfill \underline{f(x)=130\left( \cfrac{1}{\sqrt[30]{2}} \right)^x} \\\\[-0.35em] ~\dotfill\\\\ \textit{when x = 56, what is f(x)?}\qquad f(x)=130\left( \cfrac{1}{\sqrt[30]{2}} \right)^(56)\implies f(x)\approx 35.647`