Question

HELP ME PLEASE

Answer 1

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:

$P(t)=A(r)^t$

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:

$\displaystyle P(t)=A\Big(\frac{1}{2}\Big)^{t/4}$

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:

$\displaystyle 2=A\Big(\frac{1}{2}\Big)^{12/4}$

Solve for A. Simplify:

$\displaystyle 2=A\Big(\frac{1}{2}\Big)^3$

Simplify:

$\displaystyle 2=A\Big(\frac{1}{8}\Big)$

Thus, the initial mass of the sample was:

$A=16\text{ mg}$

5 weeks is equivalent to 35 days. Therefore, we can find P(35):

$\displaystyle P(35)=16\Big(\frac{1}{2}\Big)^{35/4}\approx0.0372\text{ mg}$

About 0.0372 mg will be left of the original 16 mg sample after 5 weeks.