Question

The half-life of Radium-226 is 1590 years. If a sample contains 500 mg, how many mg will remain after 4000 years?

Answer 1

for the following variation of the problem

The half-life of radium-226 is about 1,590 years. How much of a 100mg sample will be left in 500 years? Write your answer rounded to the nearest tenth.

A≈80.41mg

Explanation:

This problem requires two main steps. First, we must find the decay constant k. If we start with 100mg, at the half-life there will be 50mg remaining. We will use this information to find k. Then, we use that value of k to help us find the amount of sample that will be left in 500 years.

Identify the variables in the formula.

AA0ktA=50=100=?=1,590years=A0ekt

Substitute the values in the formula.

50=100ek⋅1,590

Solve for k. Divide each side by 100.

0.5=e1,590k

Take the natural log of each side.

ln0.5=lne1,590k

Use the power property.

ln0.5=1,590klne

Simplify.

ln0.5=1,590k

Divide each side by 1,590.

ln0.51,590=k (exact answer)

We use this rate of growth to predict the amount that will be left in 500 years.

AA0ktA=?=100=ln0.51,590=500years=A0ekt

Substitute in the values.

A=100eln0.51,590⋅500

Evaluate.

In 500 years, there will be approximately 80.4mg remaining.

Answer 2

87.43 mg

Explanation:

Let the decay rate of Radium-226 is r% annually.

So, we can write the equation of decay as
`x = P(1 - (r)/(100))^(t)` ....... (1) where, x is final amount after decay of t years and P is initial amount of radium-226.

Hence,
`(P)/(2) = P(1 - (r)/(100) )^(1590)`

⇒
`(1 - (r)/(100))^(1590) = 0.5`

⇒
`(1 - (r)/(100) ) = (0.5)^{(1)/(1590) } = 0.999564`

Now, we have to calculate the remaining amount of Radium-226 after 4000 years if the initial amount is 500 mg.

So, we can write
`x = 500(1 - (r)/(100) )^(4000) = 500(0.999564)^(4000) = 87.43` mg. (Answer)