The volume of a sphere is 48m^3 What is the volume of a cylinder if its radius is the same as the sphere's and its height is equal to the sphere's diameter ? ( 1 point )
To find the volume of a cylinder, we use the formula V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the radius of the cylinder is the same as the sphere's, we can assume that the radius of the sphere is also the radius of the cylinder.
Now, let's consider the height of the cylinder. It is given that the height is equal to the sphere's diameter. Since the diameter of a sphere is twice the radius, the height of the cylinder is 2 times the radius.
Let's represent the radius of the sphere and cylinder as r, and the height of the cylinder as 2r.
We are given that the volume of the sphere is 48m^3, so we can write the formula for the volume of a sphere as:
V_sphere = (4/3)πr^3 = 48
Now, let's calculate the volume of the cylinder using the given information:
V_cylinder = πr^2 * 2r
Since the radius of the cylinder is the same as the sphere's, we can substitute r into the formula:
V_cylinder = π * r^2 * (2r)
Simplifying the expression, we get:
V_cylinder = 2πr^3
Therefore, the volume of the cylinder is 2πr^3.
However, we need to find the exact value of the volume, so we can substitute the value of r from the volume of the sphere equation:
(4/3)πr^3 = 48
Simplifying the equation, we get:
(4/3)πr^3 = 48
Dividing both sides by (4/3)π, we get:
r^3 = 36
Taking the cube root of both sides, we get:
r = 3
Substituting this value of r into the volume of the cylinder equation:
V_cylinder = 2π(3)^3 = 2π(27) = 54π
Therefore, the volume of the cylinder is 54π m^3 or approximately 169.65 m^3.