Question

The amount of a radioactive isotope present in a certain sample at time t is given by A(t)=600e−0.02838t grams, where t years is the time since the initial amount was measured.a. Find the initial amount of the isotope that is present in the sample.b. Find the half-life of this isotope. That is, find the number of years until half of the original amount of the isotope remains.

Answer 1

The initial amount of isotope is 600. The number of years till half-life is 24.51 years

Using the expression given :

• 
  `A(t)=600e^(−0.02838t)`
• t = time

A.)

Initial Amount will occur at time, t = 0

`A(0)=600e^(−0.02838(0))`

`A(0)=600 * 1`

A(0) = 600

Hence, initial amount of the isotope is 600.

B.)

Half-life of the isotope is the time taken for the isotope to decay to half of it's original amount.

Size = Initial amount/2

• 600/2 = 300

Now we have ;

`300 = 600e^(−0.02838(t))`

`300/600 = e^(−0.02838(t))`

`(1)/(2) = e^(−0.02838(t))`

`t = (-1)/(0.02828) * In((1)/(2))`

`t = 24.51`

Answer 2

We have the following equation for the amount in grams of radiactive isotope:

`A(t)=600\cdot e^(-0.02828t)`

where t denotes the time in years.

a. Find the initial amount of the isotope that is present in the sample.

The initial amount of the isotope occurs at t=0. Then, by substituting this value into to our equation, we have

`\begin{gathered} A(0)=600\cdot e^(0.02828(0)) \\ A(0)=600\cdot1 \\ A(0)=600 \end{gathered}`

Then, the initial amount is 600 grams.

b. Find the​ half-life of this isotope.

In this case, the final amount must be half of the starting material, that is,

`A(t)=(600)/(2)=300`

and the half-life is the corresponding time for this amount of material. So, we have

`300=600\cdot e^(-0.02838\cdot t)`

By moving 600 to the left hand side, we get

`\begin{gathered} (300)/(600)=e^(-0.02838\cdot t) \\ or\text{ equivalently.} \\ e^(-0.02838\cdot t)=(1)/(2) \end{gathered}`

therefore, the time is given as

`t=-(1)/(0.02828)\ln ((1)/(2))`

then, the time is

`t=24.423\text{ years}`

then. by rounding down, the answer is 24 years.