Find the volume of the sphere if the volume of a cone with the same radius and h = 2r is 297 in^3 .
The formula for the volume of a cone is V_cone = (1/3)πr^2h, where r is the radius and h is the height. Since h = 2r in this case, the volume of the cone can be expressed as V_cone = (1/3)πr^2(2r) = (2/3)πr^3.
Given that the volume of the cone is 297 in^3, we can set up the equation:
(2/3)πr^3 = 297
Multiplying by 3/2π gives:
r^3 = 297 * 3 / 2
r^3 = 445.5
r = ∛445.5
r ≈ 7.4 inches
Now, we can calculate the volume of the sphere using the formula V_sphere = (4/3)πr^3:
V_sphere = (4/3)π(7.4)^3
V_sphere ≈ (4/3)π(405.224)
V_sphere ≈ 540.299 in^3
Therefore, the volume of the sphere is approximately 540.299 cubic inches.