Victor K. answered • 01/18/19

Perfect score on AP Calculus exam

Let's start with the formula for the volume of a cone: V = (π / 3) r^{2} 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) r^{3}

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 y^{4} would be 4y^{3} y'.

Derivative: V' = (π / 3) * (3r^{2} r' ) = π r^{2} r'

We know V' = 14 and we are given the current height = 7, which also means the current radius = 7.

So 14 = π * (7^{2}) * r'

r' = 14 / (49π)