Question

Astatine is a radioactive chemical element that was first produced at the University of California, Berkeley in 1940. The half-life of astatine is 8 hours.

Write an exponential function that models the mass in grams y remaining from a 560-gram sample after t hours.

When will approximately 198 grams remain in the sample?

How many grams will remain after 3 days? Express your answer as a decimal rounded to the nearest hundredth.

Answer 1

Approximately 198 grams will remain in the sample after 12 hours.

Approximately 1.09 grams will remain after three days.

Explanation:

We can write an exponential function to model the situation. The exponential model for decay is:

`\displaystyle A=A_0(r)^(t/h)`

Where A₀ is the initial amount, r is the rate of decay, t is the time that has passed (in this case in hours), and h is the half-life.

Since the half-life of the chemical, astatine, is 8 hours, h = 8 and r = 0.5. The initial amount is 560 grams. Hence:

`\displaystyle A=560\left((1)/(2)\right)^(t/8)`

To find when the sample will have approximately 198 grams, remaining, let A = 198 and solve for t:

`198=560(0.5)^(t/8)`

Solve for t:

`\displaystyle (198)/(560)=(99)/(280)=\left((1)/(2)\right)^(t/8)`

Take the natural log of both sides:

`\displaystyle \ln(99)/(280)=\ln\left(\left((1)/(2)\right)^(t/8)\right)`

Using logarithm properties:

`\displaystyle (t)/(8)\ln(1)/(2)=\ln(99)/(280)`

So:

`\displaystyle t=(8\ln(99/280))/(\ln(0.5))=11.9994...\approx 12\text{ hours}`

Approximately 198 grams remain in the sample after 12 hours.

Three days is equivalent to 72 hours. Hence, t = 72:

`\displaystyle A(72)=560\left((1)/(2)\right)^(72/8)=1.09375\approx 1.09\text{ grams}`

Approximately 1.09 grams of astatine will remain after three days.