carbon - 14 has a half - life of approximately 5,730 years. this exponential decay can be modeled with the…

carbon - 14 has a half - life of approximately 5,730 years. this exponential decay can be modeled with the function $n(t)=n_0(\frac{1}{2})^{\frac{t}{5730}}$. if an organism had 200 atoms of carbon - 14 at death, how many atoms will be present after 14,325 years? round the answer to the nearest hundredth. atoms

carbon - 14 has a half - life of approximately 5,730 years. this exponential decay can be modeled with the function $n(t)=n_0(\frac{1}{2})^{\frac{t}{5730}}$. if an organism had 200 atoms of carbon - 14 at death, how many atoms will be present after 14,325 years? round the answer to the nearest hundredth. atoms

Answer

Explanation:

Step1: Identify the values

$N_0 = 200$ (initial amount), $t = 14325$ (time passed), $h=5730$ (half - life).

Step2: Substitute into the formula

$N(t)=N_0\left(\frac{1}{2}\right)^{\frac{t}{h}}$, so $N(14325)=200\left(\frac{1}{2}\right)^{\frac{14325}{5730}}$.

Step3: Calculate the exponent

$\frac{14325}{5730}=2.5$.

Step4: Calculate the value of the power

$\left(\frac{1}{2}\right)^{2.5}=\frac{1}{2^{2.5}}=\frac{1}{\sqrt{2^5}}=\frac{1}{\sqrt{32}}\approx\frac{1}{5.656854}$.

Step5: Calculate the final amount

$N(14325)=200\times\frac{1}{5.656854}\approx 35.35$.

Answer:

$35.35$