an image shows three objects, a cone a cylinder and a sphere, the cone has a height of 3r and the cylinder also has a height of 3r.
the volume of the cone is 25(pi) using this information, answer the question below:
How many times larger is the volume of the sphere than the volume of the cone IF the cone and the sphere had the same dimensions?
Possible answers: A: 3R B: 100/3(PI) C: 75/2(PI) D: 75(PI) E: 1/3 F: 25(PI) G: 3 H: 4R I: 125(PI) J: 50/3(PI) K: 2
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
Given that the height of the cone is 3r and the volume of the cone is 25π, we can use the formula to find the radius of the cone:
25π = (1/3)πr^2(3r)
25π = (1/3)π(3r)(r^2)
25 = r^3
r = ∛25
r = 2.924
Now, we know that the volume of the sphere is V_sphere = (4/3)πr^3. Since the cone and the sphere have the same dimensions (same radius and height), the radius of the sphere is also 2.924.
Therefore, the volume of the sphere is:
V_sphere = (4/3)π(2.924)^3
V_sphere ≈ 108π
Now let's find how many times larger the volume of the sphere is compared to the volume of the cone:
108π / 25π ≈ 4.32
So, the volume of the sphere is 4.32 times larger than the volume of the cone.
Since none of the given answers match 4.32, we can round to the nearest whole number and choose:
H: 4R