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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:
[tex]\displaystyle A=A_0(r)^{t/h}[/tex]
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:
[tex]\displaystyle A=560\left(\frac{1}{2}\right)^{t/8}[/tex]
To find when the sample will have approximately 198 grams, remaining, let A = 198 and solve for t:
[tex]198=560(0.5)^{t/8}[/tex]
Solve for t:
[tex]\displaystyle \frac{198}{560}=\frac{99}{280}=\left(\frac{1}{2}\right)^{t/8}[/tex]
Take the natural log of both sides:
[tex]\displaystyle \ln\frac{99}{280}=\ln\left(\left(\frac{1}{2}\right)^{t/8}\right)[/tex]
Using logarithm properties:
[tex]\displaystyle \frac{t}{8}\ln\frac{1}{2}=\ln\frac{99}{280}[/tex]
So:
[tex]\displaystyle t=\frac{8\ln(99/280)}{\ln(0.5)}=11.9994...\approx 12\text{ hours}[/tex]
Approximately 198 grams remain in the sample after 12 hours.
Three days is equivalent to 72 hours. Hence, t = 72:
[tex]\displaystyle A(72)=560\left(\frac{1}{2}\right)^{72/8}=1.09375\approx 1.09\text{ grams}[/tex]
Approximately 1.09 grams of astatine will remain after three days.