Given: \(0.01<\frac{1}{3^k}<0.1\)

Rewrite as follows: \(\frac{1}{100}<\frac{1}{3^k}<\frac{1}{10}\)

At this point we can see that it must be the case that \(10 < 3^k < 100\)

Since \(k\) must be an integer, we can see that there are two possible solutions that satisfy the inequality \(10 < 3^k < 100\).
They are: \(k = 3\) and \(k = 4\)

So the product of the solutions \(= (3)(4) = 12\)

Answer: D