Question

A radioactive substance decays exponentially. A scientist begins with 140 milligrams of a radioactive substance. After 36 hours, 70 mg of the substance remains.How many milligrams will remain after 48 hours?

Answer 1

The exponential decay formula is given as,

`A=\text{ A}_0e^(-kt)`

Where,

`\begin{gathered} A_0\text{ = initial value = 140 mg} \\ A\text{ = Amount after t minutes.} \end{gathered}`

At t = 36 hours, 70 mg remains.

Therefore,

`\begin{gathered} 70\text{ = 140}*\text{ e}^(-k*36) \\ (70)/(140)\text{ = e}^(-36k) \\ (1)/(2)\text{ = e}^(-36k) \end{gathered}`

Taking log on both sides,

`\begin{gathered} ln((1)/(2))\text{ = ln\lparen e}^(-36k)) \\ ln((1)/(2))\text{ = -36k} \\ k\text{ = }(-ln(0.5))/(36) \\ k\text{ = }(-(-0.6931))/(36) \\ k\text{ = }(0.6931)/(36) \\ k\text{ = 0.0193} \\ \end{gathered}`

After 48 hours, the amount remaining is calculated as.

`\begin{gathered} A=\text{ 140 }*\text{ e}^(-0.0193*48) \\ A\text{ = 140 }*\text{ e}^(-0.9264) \\ A\text{ = 140 }*\text{ 0.3960} \\ A\text{ = 23.76 }\approx\text{ 24 milligrams.} \end{gathered}`

Thus the amount remaining after 48 hours is approximately 24 milligrams.