A cone-shaped mountain has its base on the ocean floor and height $9000$ feet. The top $\frac{1}{27}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?

1. The top cone is similar to the whole mountain. Its volume fraction is the cube of its height fraction, so $\left(\frac{h_{\text{top}}}{9000}\right)^3=\frac1{27}$ and $h_{\text{top}}=3000$ feet.
2. The ocean depth is $9000-3000=\boxed{6000}$ feet.