Question

this chart show the amount of radioactivity measured from three unknown isotopes.

isotope	starting weight	ending measured weight	half - life
b	20	2.5	2 days
c	45	22.5	10 days

which statement describes each of the isotopes?
isotope a was measured at day 24, isotope b was measured at day 6, and isotope c was measured at day 10.
isotope a was measured at day 18, isotope b was measured at day 4, and isotope c was measured at day 5.
isotope a was measured at day 24, isotope b was measured at day 8, and isotope c was measured at day 20.
isotope a was measured at day 30, isotope b was measured at day 6, and isotope c was measured at day 10.

Explanation:

Step1: Recall radioactive - decay formula

The formula for radioactive decay is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the ending amount, $N_0$ is the starting amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life.

Step2: Calculate time for Isotope A

Given $N_0 = 95$, $N = 5.9$, and $T_{1/2}=6$ days. Substitute into the formula: $5.9 = 95(\frac{1}{2})^{\frac{t}{6}}$. First, $\frac{5.9}{95}=(\frac{1}{2})^{\frac{t}{6}}$, so $0.0621 = (\frac{1}{2})^{\frac{t}{6}}$. Take the logarithm of both sides: $\log(0.0621)=\frac{t}{6}\log(\frac{1}{2})$. Then $t = 6\times\frac{\log(0.0621)}{\log(0.5)}\approx24$ days.

Step3: Calculate time for Isotope B

Given $N_0 = 20$, $N = 2.5$, and $T_{1/2}=2$ days. Substitute into the formula: $2.5 = 20(\frac{1}{2})^{\frac{t}{2}}$. First, $\frac{2.5}{20}=(\frac{1}{2})^{\frac{t}{2}}$, so $0.125 = (\frac{1}{2})^{\frac{t}{2}}$. Since $0.125=\frac{1}{8}=(\frac{1}{2})^3$, then $\frac{t}{2}=3$ and $t = 6$ days.

Step4: Calculate time for Isotope C

Given $N_0 = 45$, $N = 22.5$, and $T_{1/2}=10$ days. Substitute into the formula: $22.5 = 45(\frac{1}{2})^{\frac{t}{10}}$. First, $\frac{22.5}{45}=(\frac{1}{2})^{\frac{t}{10}}$, so $0.5 = (\frac{1}{2})^{\frac{t}{10}}$. Then $\frac{t}{10}=1$ and $t = 10$ days.

Answer:

Isotope A was measured at day 24, Isotope B was measured at day 6, and Isotope C was measured at day 10.