Radioactive carbon, ${}^{14}\mathrm{C}$ , has a has a half-life of 5730 years. After 1000 years, 3.7 grams remain. What was the initial quantity of ${}^{14}\mathrm{C}$ ? Round your answer to two decimal places.

A) 4.18 grams
B) 3.90 grams
C) 4.35 grams
D) 6.98 grams
E) 3.94 grams

The amount of radioactive carbon 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. Rearranging to solve for $A_0$ , we get $A_0=A{\left(\frac{1}{2}\right)}^{-\frac{t}{T}}$ . Substituting $A=3.7$ grams, $t=1000$ years, and $T=5730$ years, we find $A_0\approx 4.18$ grams.