Question

A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 36 grams. Write an exponential equation f(t) representing this situation. (Let f be the amount of radioactive substance in grams and t be the time in minutes.)

Answer 1

f(t) = 250
`e^(-0.007752t)`

Explanation:

Let f(t) =
`\alpha`
`e^(\beta t )`

where f is the amount of radioactive substance in grams

and t is the time in minutes

initially (at t=0), f = 250 grams

⇒f(0) = 250 grams

⇒
`\alpha`
`e^(0\beta)` = 250

⇒
`\alpha`
`e^(0)` = 250

⇒
`\alpha` = 250 grams {∵
`e^(0) = 1`}

⇒f(t) = 250
`e^(\beta t )`

At t = 250 minutes, f = 36 grams

⇒f(250) = 36 grams

⇒250
`e^(250\beta)` = 36

⇒
`e^(250\beta)` =
`(36)/(250)` = 0.144

⇒250
`\beta` = ㏑ 0.144 = -1.938

⇒
`\beta` = -
`(1.938)/(250)` = -0.007752
`min^(-1)`

∴f(t) = 250
`e^(-0.007752t)`