Let's start with the formula for the volume of a cone: V = (π / 3) r2 h
We are given that the height is always half the diameter, aka it's equal to the radius, and so h = r. We can plug this into our formula right away:
V = (π / 3) r3
For related rates problems, we take the derivative of both sides, keeping in mind that every variable we differentiate gets a derivative tagged on, not just x. For example, the derivative of y4 would be 4y3 y'.
Derivative: V' = (π / 3) * (3r2 r' ) = π r2 r'
We know V' = 14 and we are given the current height = 7, which also means the current radius = 7.
So 14 = π * (72) * r'
r' = 14 / (49π)