The radioactive isotope Bismuth-210 has a half-life of 5 days. Suppose we have an initial amount of 100 mg. The amount of Bismuth-210 remaining after $t$ days is
A) $100 \left( 2 ^ { 0.2 t } \right)$
B) $5 \left( 2 ^ { 0.2 t } \right)$
C) $50 \left( 2 ^ { - 0.2 t } \right)$
D) $100 \left( 2 ^ { - 0.2 t } \right)$
E) $100 \left( e ^ { 0.2 t } \right)$
F) $5 \left( e ^ { 0.2 t } \right)$
G) $50 \left( e ^ { - 0.2 t } \right)$
H) $100 \left( e ^ { - 0.2 t } \right)$

Question

The radioactive isotope Bismuth-210 has a half-life of 5 days. Suppose we have an initial amount of 100 mg. The amount of Bismuth-210 remaining after $t$ days is
A) $100 \left( 2 ^ { 0.2 t } \right)$ 
B) $5 \left( 2 ^ { 0.2 t } \right)$ 
C) $50 \left( 2 ^ { - 0.2 t } \right)$ 
D) $100 \left( 2 ^ { - 0.2 t } \right)$ 
E) $100 \left( e ^ { 0.2 t } \right)$ 
F) $5 \left( e ^ { 0.2 t } \right)$ 
G) $50 \left( e ^ { - 0.2 t } \right)$ 
H) $100 \left( e ^ { - 0.2 t } \right)$

Quizplus · Accepted Answer

The formula for the decay of a radioactive substance is $A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}$, where $A$ is the amount remaining, $A_0$ is the initial amount, $t$ is the time elapsed, and $T$ is the half-life. Substituting $A_0 = 100$, $T = 5$, and converting the base from $2$ to $e$ is unnecessary here. The correct formula simplifies to $100 \left( 2 ^ { - 0.2 t } \right)$, matching option D.

The Radioactive Isotope Bismuth-210 Has a Half-Life of 5 Days ttt

The Radioactive Isotope Bismuth-210 Has a Half-Life of 5 Days $t$