corn is being poured onto a conical pile at the rate of 14ft^3/min. in the pile the altitude is always half of the diameter. find the rate the radius is changing when the height is 7ft

Answer 1

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)