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For this question, is this interpretation entirely correct? Why couldn't we directly use $V = \frac{\pi}{3} \times r^3$, where $r = 6$ to get us the solution?

I'm not entirely sure what the image attached is doing?

Thanks

Edit: having trouble formatting so my picture appears in the post, sorry.

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    The 2nd diagram is showing you that the problem can be simplified into a cross-section. It shows the cylinder inside the sphere as a cross-section taken through the North and South poles. – 2017-02-18

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Yes, the given solution is correct. But we can't use volume of the sphere.

The thing is, we need to find the volume of the cylinder and not the volume of the sphere. If it were given that we need to find the volume of cylinder in terms of the volume of the sphere, then we'd use that formula.

But, we instead use $V=\pi \times r^2 \times h$ and replace $r$ in terms of $h$ in the first case and replace $h$ in terms of $r$ in the second case.

Hope this helps.