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Math  /  Algebra

QuestionComplete the table shown to the right for the half-life of a certain radioactive substance. \begin{tabular}{|l|l|} \hline Half-Life & Decay Rate, k\mathbf{k} \\ \hline 3851 years & \\ \hline \end{tabular} k=k= □\square (Round to six decimal places as needed.)

Studdy Solution

STEP 1

1. The decay of a radioactive substance follows the exponential decay model.
2. The relationship between half-life and decay rate is given by the formula k=ln⁡(2)t12 k = \frac{\ln(2)}{t_{\frac{1}{2}}} , where t12 t_{\frac{1}{2}} is the half-life.

STEP 2

1. Identify the formula for decay rate.
2. Substitute the given half-life into the formula.
3. Calculate the decay rate k k and round to six decimal places.

STEP 3

Identify the formula for the decay rate k k in terms of the half-life t12 t_{\frac{1}{2}} :
k=ln⁡(2)t12 k = \frac{\ln(2)}{t_{\frac{1}{2}}}

STEP 4

Substitute the given half-life of 3851 years into the formula:
k=ln⁡(2)3851 k = \frac{\ln(2)}{3851}

STEP 5

Calculate k k using the natural logarithm of 2, which is approximately 0.693147 0.693147 :
k=0.6931473851 k = \frac{0.693147}{3851}
Perform the division to find k k :
k≈0.0001799 k \approx 0.0001799
Round k k to six decimal places:
k≈0.0001799 k \approx 0.0001799
The decay rate k k is:
0.0001799 \boxed{0.0001799}

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