Question

the amount of a sample remaining after t days is given by the equation $p(t)=a(\frac{1}{2})^{\frac{t}{h}}$, where a is the initial amount of the sample and h is the half - life, in days, of the substance. a scientist has a 10 - mg sample of a radioactive isotope. the isotope has a half - life of 8 days. after 16 days, how much of the radioactive isotope remains?
2.0 mg
2.5 mg
5.7 mg
7.1 mg

Explanation:

Step1: Identify values

Given $A = 10$ mg (initial amount), $h=8$ days (half - life), $t = 16$ days (time passed).

Step2: Substitute into formula

The formula is $P(t)=A(\frac{1}{2})^{\frac{t}{h}}$. Substitute $A = 10$, $h = 8$ and $t=16$ into it: $P(16)=10\times(\frac{1}{2})^{\frac{16}{8}}$.

Step3: Calculate exponent

First, calculate $\frac{16}{8}=2$. So $P(16)=10\times(\frac{1}{2})^{2}$.

Step4: Calculate power

$(\frac{1}{2})^{2}=\frac{1}{4}$. Then $P(16)=10\times\frac{1}{4}$.

Step5: Final calculation

$10\times\frac{1}{4}=\frac{10}{4}=2.5$ mg.

Answer:

2.5 mg