The height, h, of a cylinder is 3 times its radius, r. Which of the following represents the rate of change of the volume, V, of the cylinder with respect to its height, h?
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To find the rate of change of the volume, V, of a cylinder with respect to its height, h, we need to differentiate the formula for the volume of a cylinder with respect to h.
The volume, V, of a cylinder is given by the formula V = πr^2h, where r is the radius of the cylinder and h is the height.
Given that the height, h, of the cylinder is 3 times its radius, r, we can substitute h = 3r into the formula for the volume: V = πr^2(3r).
Simplifying this equation, we get V = 3πr^3.
To find the rate of change of the volume with respect to the height, we differentiate this equation with respect to h.
dV/dh = d/dh(3πr^3)
Since r is a constant with respect to h, its derivative is zero. Therefore, we can simplify the equation to:
dV/dh = 0
So, the rate of change of the volume, V, of the cylinder with respect to its height, h, is 0.