Volume of Cones, Cylinders, and Spheres Quick Check%0D%0A3 of 53 of 5 Items%0D%0A%0D%0AQuestion%0D%0AThe 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)%0D%0AResponses%0D%0A%0D%0A252 π cm3%0D%0AStart Fraction 25 over 2 End Fraction pi cm cubed%0D%0A%0D%0A503 π cm3%0D%0AStart Fraction 50 over 3 End Fraction pi cm cubed%0D%0A%0D%0A25 π cm3%0D%0A25 pi cm cubed%0D%0A%0D%0A256 π cm3
The correct answer is 503 π cm3.
To find the volume of the sphere, we can use the formula for the volume of a cone, which is V = (1/3)πr^2h, where r is the radius and h is the height.
Since the height of the cone is equal to the sphere's diameter, we can write h = 2r.
Plugging this into the formula, we get V = (1/3)πr^2(2r) = (2/3)πr^3.
We are given that the volume of the cone is 253 π cm3, so we can set up the equation (2/3)πr^3 = 253 π.
Divide both sides by (2/3)π to solve for r^3:
r^3 = (253 π) / ((2/3)π) = (253/2)(3) = 379.5
Now, take the cube root of both sides to solve for r:
r = (379.5)^(1/3) ≈ 7.44
Finally, plug this value of r into the formula for the volume of a sphere, V = (4/3)πr^3, to find the volume of the sphere:
V = (4/3)π(7.44)^3 ≈ 503 π cm3.
Hence, the volume of the sphere is 503 π cm3.