Just in case a picture helps...

Related rates nearly always depend on the chain rule, so you might want to try filling up this pattern...

... where straight continuous lines differentiate downwards (integrate up) with respect to the main variable (usually time), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

So what have we got here? V, dV/dt, a choice of either r or h because they depend on each other, and then dh/dt the rate of increase in h. So it looks as though we should get r in terms of h rather than the other way round.

r/h = 5/22 so r = 5h/22 so the volume formula in terms of h is...

So differentiate with respect to the inner function, and the inner function with respect to t, and sub in the given values of h and dh/dt...

Spoiler:

Finally, of course, add the 12800.

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Don't integrate - balloontegrate!

Balloon Calculus: Gallery

Balloon Calculus Drawing with LaTeX and Asymptote!