the radius r of a right circular cone is increasing at a rate of 2cm/min.the height h of the cone is related to the radius by h=3r.find the rate of change of the volume when r=6cm

v = 1/3 π r^2 h = 1/3 π r^2 (3r) = πr^3

dv/dt = 3πr^2 dr/dt

now plug in your numbers.

ican do the question please help me

To find the rate of change of the volume of the cone, we need to use the formula for the volume of a cone and differentiate it with respect to time.

The volume of a cone can be given by the formula:

V = 1/3 * π * r^2 * h

where V is the volume, r is the radius, and h is the height.

We are given that the height h is related to the radius r by the equation h = 3r.

Let's differentiate the volume formula with respect to time (t):

dV/dt = 1/3 * π * (2r * dr/dt * h) + 1/3 * π * r^2 * dh/dt

Since we are given that dr/dt = 2 cm/min and dh/dt = 0 (as there is no information given about the rate of change of height), we can substitute these values into the equation:

dV/dt = 1/3 * π * (2r * 2) + 0

Simplifying this gives:

dV/dt = 4/3 * π * r

Now, we can substitute r = 6 cm into the equation to find the rate of change of the volume:

dV/dt = 4/3 * π * 6
dV/dt = 24π cm^3/min

Therefore, when the radius is 6 cm, the rate of change of the volume is 24π cm^3/min.

To find the rate of change of the volume, we need to differentiate the volume formula with respect to time.

The volume of a right circular cone can be calculated using the formula V = (1/3)πr^2h.

Given that h = 3r, we can substitute this into the volume formula to eliminate h:

V = (1/3)πr^2(3r)
V = πr^3

Now, we need to differentiate this volume formula with respect to time t, since the radius is changing with time.

dV/dt = d/dt(πr^3)

To differentiate this equation, we need to use the chain rule:

dV/dt = 3πr^2(dr/dt)

Now, we can substitute the given information dr/dt = 2 cm/min and r = 6 cm:

dV/dt = 3π(6 cm)^2(2 cm/min)
dV/dt = 3π(36 cm^2)(2 cm/min)
dV/dt = 216π cm^3/min

Thus, the rate of change of the volume when r = 6 cm is 216π cm^3/min.