The isotope ${ } ^ { 14 } \mathrm { C }$ has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to four decimal places.
A) $7.2469 \mathrm { gm } , 4.1988 \mathrm { gm }$
B) $6.2117 \mathrm { gm } , 3.5989 \mathrm { gm }$
C) $10.3528 \mathrm { gm } , 5.9982 \mathrm { gm }$
D) $4.1411 \mathrm { gm } , 2.3993 \mathrm { gm }$
E) $12.4233 \mathrm { gm } , 7.1979 \mathrm { gm }$

Question

Quizplus · Accepted Answer

The amount of a radioactive isotope remaining after a certain period can be calculated using the formula $A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}$, where $A$ is the amount remaining, $A_0$ is the initial amount, $t$ is the time elapsed, and $T$ is the half-life of the isotope. For ${ } ^ { 14 } \mathrm { C }$ with a half-life of 5,715 years, after 500 years and 5,000 years, the calculations are as follows:After 500 years:$A = 11 \left( \frac{1}{2} \right)^{\frac{500}{5715}} \approx 10.3528 \, \text{gm}$After 5,000 years:$A = 11 \left( \frac{1}{2} \right)^{\frac{5000}{5715}} \approx 5.9982 \, \text{gm}$Therefore, the correct answer is $10.3528 \, \text{gm}$ after 500 years and $5.9982 \, \text{gm}$ after 5,000 years.

The Isotope 14C{ } ^ { 14 } \mathrm { C }14C

The Isotope ${ } ^ { 14 } \mathrm { C }$