Question

If the diameter of a right circular cylinder is equal to its height, then the volume of the cylinder, in terms of r, will be ?
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Answer 1

Answer:

The formula for the volume of a right circular cylinder is V = πr^2h, where r is the volume and h is the height of the cylinder.

The formula for the volume of a sphere is V = 4πr^3/3, where r is the radius of the sphere.

Let r be the radius of the cylinder, h be the height of the cylinder, and R be the radius of the sphere.

Since the volumes of the cylinder and the sphere are equal:

π(r^2)h = 4π(R^3)/3.

So (r^2)h = 4(R^3)/3.

Since the radius of the cylinder and the diameter of the sphere are equal:

r = 2R.

So R = r/2.

So (r^2)h = 4[(r/2)^3]/3.

So (r^2)h = 4[(r^3)/(2^3)]/3.

So (r^2)h = [4(r^3)/8]/3.

So (r^2)h = [(r^3)/2]/3.

So (r^2)h = (r^3)/(2×3).

So (r^2)h = (r^3)/6.

So (r^2)h = r(r^2)/6.

So either:

(A) r^2 = 0.

So r = 0.

So R = r/2 = 0/2 = 0.

Or

(B) h = r/6.

So h/r = 1/6.

So either:

(A) The radius of both the cylinder and the sphere are zero, and so the volume of both the cylinder and the sphere are zero.

Or

(B) The ratio of the height of the cylinder to its radius is 1/6. That is, the height of the cylinder is one sixth of its radius.

CHECK:

If h = r/6, then the volume of the cylinder is π(r^2)h = π(r^2)(r/6) = π(r^3)/6.

Since R = r/2, the volume of the sphere is 4π(R^3)/3 = 4π[(r/2)^3]/3 = [4π(r^3)/(2^3)]/3 = [4π(r^3)/8]/3 = [π(r^3)/2]/3 = π(r^3)/(2×3) = π(r^3)/6.

So the volumes of the cylinder and the sphere are equal, as desired.

Step-by-step explanation:

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Answer 2

Answer:

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Step-by-step explanation:

The formula for the volume of a right circular cylinder is V = πr^2h, where r is the volume and h is the height of the cylinder.

The formula for the volume of a sphere is V = 4πr^3/3, where r is the radius of the sphere.

Let r be the radius of the cylinder, h be the height of the cylinder, and R be the radius of the sphere.

Since the volumes of the cylinder and the sphere are equal:

π(r^2)h = 4π(R^3)/3.

So (r^2)h = 4(R^3)/3.

Since the radius of the cylinder and the diameter of the sphere are equal:

r = 2R.

So R = r/2.

So (r^2)h = 4[(r/2)^3]/3.

So (r^2)h = 4[(r^3)/(2^3)]/3.

So (r^2)h = [4(r^3)/8]/3.

So (r^2)h = [(r^3)/2]/3.

So (r^2)h = (r^3)/(2×3).

So (r^2)h = (r^3)/6.

So (r^2)h = r(r^2)/6.

So either:

(A) r^2 = 0.

So r = 0.

So R = r/2 = 0/2 = 0.

Or

(B) h = r/6.

So h/r = 1/6.

So either:

(A) The radius of both the cylinder and the sphere are zero, and so the volume of both the cylinder and the sphere are zero.

Or

(B) The ratio of the height of the cylinder to its radius is 1/6. That is, the height of the cylinder is one sixth of its radius.

CHECK:

If h = r/6, then the volume of the cylinder is π(r^2)h = π(r^2)(r/6) = π(r^3)/6.

Since R = r/2, the volume of the sphere is 4π(R^3)/3 = 4π[(r/2)^3]/3 = [4π(r^3)/(2^3)]/3 = [4π(r^3)/8]/3 = [π(r^3)/2]/3 = π(r^3)/(2×3) = π(r^3)/6.

So the volumes of the cylinder and the sphere are equal, as desired.

Step-by-step explanation:

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