The volume of a sphere is 72 m^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?
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where V is the volume and r is the radius.
Given that the volume of the sphere is 72 m^3, we can set up the equation as follows:
72 = (4/3)πr^3
To find the volume of the cone, we can use the formula V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the radius of the cone is the same as the radius of the sphere (r) and the height of the cone is equal to the diameter of the sphere (2r), we can substitute these values into the formula:
V = (1/3)πr^2(2r)
Multiplying the terms inside the parentheses:
V = (2/3)πr^3
We know that the volume of the sphere (72 m^3) is equal to the volume of the cone, so we can set up the equation:
72 = (2/3)πr^3
To solve for the volume of the cone, we need to isolate r^3 by dividing both sides of the equation by (2/3)π:
r^3 = (72 * 3) / (2π)
r^3 = 108 / π
Now, we can find the cube root of both sides to solve for the radius of the cone:
r = (108 / π)^(1/3)
r ≈ 3.245 meters
Now that we know the radius of the cone, we can use the formula V = (1/3)πr^2h to calculate the volume of the cone.
Given that the height of the cone is equal to the diameter of the sphere (2r), we can substitute these values into the formula:
V = (1/3)π(3.245)^2(2 * 3.245)
V ≈ 72 m^3
Therefore, the volume of the cone is approximately equal to 72 m^3.