The volume of a cone is 253 π cm3. What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
25 π cm3
The correct answer is 503 π cm3.
To find the volume of the sphere, we need to first find the radius of the cone. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the volume of the cone is 253 π cm3 and the height is equal to the diameter of the sphere, we can write:
253 π = (1/3)πr^2(2r)
Simplifying this equation, we get:
253 = (2/3)r^3
Dividing both sides by (2/3), we get:
r^3 = 253 * (3/2)
r^3 = 380.5
Taking the cube root of both sides, we get:
r = ∛380.5
Now that we know the radius of the sphere, we can find its volume using the formula V = (4/3)πr^3:
V = (4/3)π(∛380.5)^3
V = (4/3)π(7 ∛5)
Simplifying this expression, we get:
V = (4/3)π * 343 ∛5
V = 428.666... ∛5
V ≈ 503.91 π cm3
Rounded to the nearest whole number, the volume of the sphere is 504 π cm3, which is equivalent to 503 π cm3.