Explanation:
Answer:
The initial mass of the sample was 16 mg.
The mass after 5 weeks will be about 0.0372 mg.
Explanation:
We can write an exponential function to model the situation.
Let the initial amount be A. The standard exponential function is given by:
P(t)=A(r)^tP(t)=A(r)
t
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:
\displaystyle P(t)=A\Big(\frac{1}{2}\Big)^{t/4}P(t)=A(
2
1
)
t/4
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:
\displaystyle 2=A\Big(\frac{1}{2}\Big)^{12/4}2=A(
2
1
)
12/4
Solve for A. Simplify:
\displaystyle 2=A\Big(\frac{1}{2}\Big)^32=A(
2
1
)
3
Simplify:
\displaystyle 2=A\Big(\frac{1}{8}\Big)2=A(
8
1
)
Thus, the initial mass of the sample was:
A=16\text{ mg}A=16 mg
5 weeks is equivalent to 35 days. Therefore, we can find P(35):
\displaystyle P(35)=16\Big(\frac{1}{2}\Big)^{35/4}\approx0.0372\text{ mg}P(35)=16(
2
1
)
35/4
≈0.0372 mg
About 0.0372 mg will be left of the original 16 mg sample after 5 weeks.