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

Answer:

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

Approximately 1.09 grams will remain after three days.

Step-by-step 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:

$\displaystyle A=560\left(\frac{1}{2}\right)^{t/8}$

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:

$\displaystyle \frac{198}{560}=\frac{99}{280}=\left(\frac{1}{2}\right)^{t/8}$

Take the natural log of both sides:

$\displaystyle \ln\frac{99}{280}=\ln\left(\left(\frac{1}{2}\right)^{t/8}\right)$

Using logarithm properties:

$\displaystyle \frac{t}{8}\ln\frac{1}{2}=\ln\frac{99}{280}$

So:

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

Approximately 198 grams remain in the sample after 12 hours.

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

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

Approximately 1.09 grams of astatine will remain after three days.