The amount of a radioactive isotope present in a certain sample at time $t$ is given by $A(t)=600 e^{-0.02831 t}$ grams, where $t$ years is the time since the initial arnount was measured. How long will it take for the amount of the isotope to equal 387 grams?
It will take about $\square$ years for the amount of the isotope to equal 387 grams.
(Do not round until the final answer. Then round to the nearest whole number as needed)
Answer
Final Answer: It will take about \(\boxed{15}\) years for the amount of the isotope to equal 387 grams.
Step 1 ：The problem is asking for the time it will take for the amount of the isotope to decrease to 387 grams. This is a problem of solving for \(t\) in the equation \(A(t)=600 e^{-0.02831 t} = 387\).
Step 2 ：We can solve this equation by taking the natural logarithm on both sides and then isolating \(t\).
Step 3 ：Let \(A_t = 387\), \(A_0 = 600\), and \(k = -0.02831\).
Step 4 ：Solving for \(t\), we get \(t = 15.489401702096947\).
Step 5 ：Rounding to the nearest whole number, we get \(t = 15\).
Step 6 ：Final Answer: It will take about \(\boxed{15}\) years for the amount of the isotope to equal 387 grams.
