Question

At the beginning of an experiment, a scientist has 120 grams of radioactive goo. After 135 minutes, her sample has decayed to 3.75 grams. Find an exponential formula for G ( t ) G(t) , the amount of goo remaining at time t t .

Answer 1

`G(t) = 120e^(-0.0257t)`

Explanation:

Amount of substance:

The amount of the substance after t minutes is given by:

`G(t) = G(0)e^(-kt)`

In which G(0) is the initial amount and k is the decay rate.

At the beginning of an experiment, a scientist has 120 grams of radioactive goo.

This means that
`G(0) = 120`, so:

`G(t) = G(0)e^(-kt)`

`G(t) = 120e^(-kt)`

After 135 minutes, her sample has decayed to 3.75 grams.

This means that
`G(135) = 3.75`.

We use this to find k. So

`G(t) = 120e^(-kt)`

`3.75 = 120e^(-135k)`

`e^(-135k) = (3.75)/(120)`

`\ln{e^(-135k)} = \ln{(3.75)/(120)}`

`-135k = \ln{(3.75)/(120)}`

`k = -\frac{\ln{(3.75)/(120)}}{135}`

`k = 0.0257`

So

`G(t) = 120e^(-0.0257t)`