The volume of a sphere is 72m ^ 3 What is the volume of a cone if its radius is the same as the sphere's and its height is equal to the sphere's diameter? (1 point)

3 answers

The volume of a sphere is given by the formula:

V_sphere = (4/3) * π * r^3

Where r is the radius of the sphere.

Since the volume of the sphere is given as 72m^3, we can set up the equation:

72 = (4/3) * π * r^3

To find the radius of the sphere, we can solve for r:

r^3 = (3/4) * (72/π)
r = (3/4) * (72/π)^(1/3)

Now, since the radius of the cone is the same as the sphere's, we can substitute this value of r into the formula for the volume of a cone:

V_cone = (1/3) * π * r^2 * h

The height of the cone is equal to the sphere's diameter, which is twice the radius, so h = 2r.

Plugging in the values:

V_cone = (1/3) * π * [(3/4) * (72/π)^(1/3)]^2 * 2 * [(3/4) * (72/π)^(1/3)]

Simplifying, we get:

V_cone = (1/3) * π * (3^2/4^2) * (72/π)^(2/3) * 2 * (3/4) * (72/π)^(1/3)

V_cone = (2/3) * π * (9/16) * (72/π)^(2/3) * (3/4) * (72/π)^(1/3)

V_cone = π * (2/3) * (9/16) * (72/π)^(2/3) * (3/4) * (72/π)^(1/3)

V_cone = (2/3) * (9/16) * (72/π)^(2/3) * (3/4) * (72/π)^(1/3)

V_cone = (2/3) * (3/4) * 9 * 72^(2/3) * 72^(1/3) / 16π

V_cone = (2/3) * (3/4) * 9 * (72 * 72)^(1/3) / 16π

V_cone = (2/3) * (3/4) * 9 * 6^(1/3) / 16π

V_cone = (2/3) * (3/4) * 9 * 2 / 16π

V_cone = (9/4) * 36 / 16π

V_cone = 9/4 * 9 / π

V_cone = 81 / 4π

Therefore, the volume of the cone is 81/4π cubic units.
sphere: 4/3 πr^3
cone: 1/3 πr^2 h = 1/3 πr^2(2r) = 2/3 πr^3
so the cone's volume is 1/2 that of the sphere: 36 m^3
Apologies for the mistake in my previous response. You are correct.

Given that the volume of the sphere is 72 m^3, the volume of the cone can be found using the formula for the volume of a cone:

V_cone = (2/3) * π * r^3

Since the radius of the cone is the same as the sphere's, we can substitute this value into the formula:

V_cone = (2/3) * π * (r^3)

V_cone = (2/3) * π * ((4/3 πr^3) / π)

Simplifying, we find:

V_cone = (2/3) * π * (4/3 * r^3) / π

V_cone = (2/3) * (4/3) * r^3

V_cone = (8/9) * (4/3) * r^3

V_cone = 32/27 * r^3

Since the radius of the cone is equal to the radius of the sphere, and the volume of the sphere is 72 m^3, we can substitute the value of r:

V_cone = 32/27 * (r^3)

V_cone = 32/27 * (72)

V_cone = 32 * 72 / 27

V_cone = 2304 / 27

V_cone ≈ 85.333 m^3

Therefore, the volume of the cone with radius equal to the sphere's and height equal to the sphere's diameter is approximately 85.333 m^3.