The volume of a cone of radius r and height h is given by v=1/3πr2h. If the radius and height are both increasing at a constant rate of 1/2 centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters.

Solution:

Given, height of the cone is 9 cm

Radius of the cone is 6 cm.

Radius and height is increasing at a constant rate of 1/2 cm/s.

We have to find the increasing volume.

We know, volume of the cone = v = 1/3πr²h

dr/dt = 0.5 cm/s

dh/dt = 0.5 cm/s

On differentiating volume,

dV/dt = (1/3)π(r²dh/dt + 2rhdr/dt)

dV/dt = (1/3)π((6)²(0.5) + 2(6)(9)(0.5))

= (1/3)π(18 + 54)

= 24π cm²/s

Therefore, the volume of the cone is increasing at the rate of 24π cm²/s.

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The volume of a cone of radius r and height h is given by v = 1/3πr²h. If the radius and height are both increasing at a constant rate of 1/2 centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters.

Summary:

The volume of a cone of radius r and height h is given by v = 1/3πr²h. If the radius and height are both increasing at a constant rate of 1/2 centimeter per second, the volume increases at the rate of 24π, in cubic centimeters per second, when the height is 9 centimeters and the radius is 6 centimeters.