Question

The radius r and height h of a right circular cone are both increasing at a constant rate of 2 cm/h. At what rate in centimeters cubed per hour is the volume V of the cone increasing when r = 6cm and h = 15cm? (The volume V of a right circular cone of height h and radius r is V = 1πr2h. )

Answer 1

The volume of the cone is increasing at a rate of 96π cm³/h.

Step-by-step explanation:

To find the rate at which the volume of the cone is increasing, we need to differentiate the volume formula V = πr^2h with respect to time and then substitute the given values.

Let's differentiate V with respect to t:

dV/dt = π(2r)(dh/dt) + π(r^2)(d/dt)(h)

Since h and r are both increasing at a constant rate of 2 cm/h, dh/dt = 2 and dr/dt = 2.

Substituting the values into the equation, we get:

dV/dt = π(2(6))(2) + π((6)^2)(2) = 24π + 72π

Therefore, the volume of the cone is increasing at a rate of 96π cm³/h when r = 6cm and h = 15cm.