65.9k views
2 votes
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.

The amount of a radioactive isotope present in a certain sample at time t is given-example-1

2 Answers

0 votes

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

User Walle Cyril
by
8.6k points
4 votes

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.

User Santa Zhang
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.