Math  /  Algebra

QuestionClick here to watch the video. A scientist determined that the bones from a mastodon had lost 83.8%83.8 \% of their carbon-14. How old were the bones at the time they were discovered? \qquad \square

Studdy Solution
Solve for the time variable to find the age of the bones.
Using the decay formula:
0.162×N0=N0×(12)t5730 0.162 \times N_0 = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{5730}}
Divide both sides by N0 N_0 :
0.162=(12)t5730 0.162 = \left(\frac{1}{2}\right)^{\frac{t}{5730}}
Take the natural logarithm of both sides to solve for t t :
ln(0.162)=ln((12)t5730) \ln(0.162) = \ln\left(\left(\frac{1}{2}\right)^{\frac{t}{5730}}\right)
ln(0.162)=t5730ln(12) \ln(0.162) = \frac{t}{5730} \cdot \ln\left(\frac{1}{2}\right)
Solve for t t :
t=ln(0.162)ln(12)×5730 t = \frac{\ln(0.162)}{\ln\left(\frac{1}{2}\right)} \times 5730
Calculate t t :
t1.8170.693×5730 t \approx \frac{-1.817}{-0.693} \times 5730
t2.622×5730 t \approx 2.622 \times 5730
t15022.26 t \approx 15022.26
The age of the bones is approximately:
15022 \boxed{15022} years

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