Asked by Victoria Cuevas on Mar 10, 2024

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Radioactive carbon, 14C{ } ^ { 14 } \mathrm { C }14C , has a has a half-life of 5730 years. After 1000 years, 3.7 grams remain. What was the initial quantity of 14C{ } ^ { 14 } \mathrm { C }14C ? 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

Half-Life

Half-life is a term used in nuclear physics and chemistry to describe the time required for one half of the atoms of a radioactive substance to undergo decay.

  • Apply the exponential model to solve real-world problems related to finance and physical sciences.
  • Gain proficiency in the application of exponential growth and decay concepts in multiple contexts, encompassing population growth and the diminishment of radioactivity.
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LL
Lindsays LaFontaine

Mar 10, 2024

Final Answer :
A
Explanation :
The amount of radioactive carbon remaining after a certain period can be calculated using the formula A=A0(12)tTA = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}A=A0(21)Tt , where AAA is the amount remaining, A0A_0A0 is the initial amount, ttt is the time elapsed, and TTT is the half-life. Rearranging to solve for A0A_0A0 , we get A0=A(12)−tTA_0 = A \left(\frac{1}{2}\right)^{-\frac{t}{T}}A0=A(21)Tt . Substituting A=3.7A = 3.7A=3.7 grams, t=1000t = 1000t=1000 years, and T=5730T = 5730T=5730 years, we find A0≈4.18A_0 \approx 4.18A04.18 grams.