Answer:
The initial mass of the sample was 16 mg.
The mass after 5 weeks will be about 0.0372 mg.
Step-by-step explanation:
We can write an exponential function to model the situation.
Let the initial amount be A. The standard exponential function is given by:
[tex]P(t)=A(r)^t[/tex]
Where r is the rate of growth/decay.
Since the half-life of Palladium-100 is four days, r = 1/2. We will also substitute t/4 for t to to represent one cycle every four days. Therefore:
[tex]\displaystyle P(t)=A\Big(\frac{1}{2}\Big)^{t/4}[/tex]
After 12 days, a sample of Palladium-100 has been reduced to a mass of two milligrams.
Therefore, when x = 12, P(x) = 2. By substitution:
[tex]\displaystyle 2=A\Big(\frac{1}{2}\Big)^{12/4}[/tex]
Solve for A. Simplify:
[tex]\displaystyle 2=A\Big(\frac{1}{2}\Big)^3[/tex]
Simplify:
[tex]\displaystyle 2=A\Big(\frac{1}{8}\Big)[/tex]
Thus, the initial mass of the sample was:
[tex]A=16\text{ mg}[/tex]
5 weeks is equivalent to 35 days. Therefore, we can find P(35):
[tex]\displaystyle P(35)=16\Big(\frac{1}{2}\Big)^{35/4}\approx0.0372\text{ mg}[/tex]
About 0.0372 mg will be left of the original 16 mg sample after 5 weeks.
