Question 2 of 10
Plutonium-240 decays according to the function $Q(t)=Q_{0} e^{-k t}$ where Q represents the quantity remaining after $t$ years and $k$ is the decay constant, $0.00011 \ldots$ To the nearest 10 years, how long will it take 27 grams of plutonium-240 to decay to 9 grams?
A. 9,990 years
B. 2,100 years
C. 1.44 years
D. 18,900 years

Step 1 ：We are given the initial quantity \(Q_0 = 27\) grams and the final quantity \(Q = 9\) grams. We are also given the decay constant \(k = 0.00011\). We need to find the time \(t\) it takes for the quantity to decay from 27 grams to 9 grams.

Step 2 ：We can do this by rearranging the decay function to solve for \(t\): \[t = \frac{1}{k} \ln \left( \frac{Q_0}{Q} \right)\]

Step 3 ：We can then substitute the given values into this equation to find \(t\).

Step 4 ：Substituting the given values, we get \(t = 9987.38444243736\)

Step 5 ：Rounding to the nearest 10 years, we get \(t = 9990.0\) years

Step 6 ：Final Answer: The time it will take for 27 grams of plutonium-240 to decay to 9 grams, to the nearest 10 years, is \(\boxed{9990}\) years.