Question

The half-life of a newly discovered radioactive element is 30 seconds. To the nearest tenth of a second, how long will it take for a sample of 9 grams to decay to 0.72 grams

Answer 1

Answer:

It will take about 109.3 seconds for nine grams of the element to decay to 0.72 grams.

Step-by-step explanation:

We can write a half-life function to model our function.

A half-life function has the form:

$\displaystyle A=A_0\left(\frac{1}{2}\right)^{t/d}$

Where A₀ is the initial amount, t is the time that has passes (in this case seconds), d is the half-life, and A is the amount after t seconds.

Since the half-life of the element is 30 seconds, d = 30. Our initial sample has nine grams, so A₀ is 9. Substitute:

$\displaystyle A=9\left(\frac{1}{2}\right)^{t/30}$

We want to find the time it will take for the element to decay to 0.72 grams. So, we can let A = 0.72 and solve for t:

$\displaystyle 0.72=9\left(\frac{1}{2}\right)^{t/30}$

Divide both sides by 9:

$\displaystyle 0.08=\left(\frac{1}{2}\right)^{t/30}$

We can take the natural log of both sides:

$\displaystyle \ln(0.08)=\ln\left(\left(\frac{1}{2}\right)^{t/30}\right)$

By logarithm properties:

$\displaystyle \ln(0.08)=\frac{t}{30}\ln(0.5)$

Solve for t:

$\displaystyle t=\frac{30\ln(0.08)}{\ln(0.5)}\approx109.3\text{ seconds}$

So, it will take about 109.3 seconds for nine grams of the element to decay to 0.72 grams.