The volume of a sphere is 185π cm3 . 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)
Responses
125π cm3
Start Fraction 12 over 5 End Fraction pi cm cubed
65π cm3
Start Fraction 6 over 5 End Fraction pi cm cubed
95π cm3
Start Fraction 9 over 5 End Fraction pi cm cubed
275π cm3
To find the volume of the cylinder, we need to find the radius and height of the cylinder.
We know that the radius of the cylinder is the same as the sphere's radius, and the height of the cylinder is equal to the sphere's diameter.
If the volume of the sphere is 185π cm^3, we can find the radius using the formula for the volume of a sphere:
V = (4/3)πr^3
185π = (4/3)πr^3
To isolate r^3, we can divide both sides by (4/3)π:
r^3 = (3/4)(185π)
r^3 = 555π/4
To find r, we can take the cube root of both sides:
r = ∛(555π/4)
Now that we have the radius, we can find the height of the cylinder, which is equal to the diameter of the sphere:
height = 2r
Substituting the value of r, we get:
height = 2(∛(555π/4))
To find the volume of the cylinder, we use the formula:
V = πr^2h
Substituting the values of r and h, we get:
V = π(∛(555π/4))^2 * 2(∛(555π/4))
Simplifying, we get:
V = π(∛555π/4)^2 * 2∛(555π/4)
V = π(∛(555π)^2/∛4)^2 * 2∛(555π/4)
V = π(555π)^2/∛4^2 * 2∛(555π/4)
V = (555π)^2/4 * 2∛(555π/4)
V = (555π)^2/4 * 2 * (555π/4)^(1/3)
V = (555π)^2 * 2^(1/3) / 4 * (555π/4)^(1/3)
Simplifying further, we get:
V = 555^2 * π * 2^(1/3) / 4 * (555/4)^(1/3)
V = 555 * π * 2^(1/3) / 4 * (555/4)^(1/3)
V ≈ 274.58π cm^3
So, the volume of the cylinder is approximately 274.58π cm^3.
Therefore, the correct option is 275π cm^3.