Question 3: The volume of a cone is 25/3 pi cubic centimeters. 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?
25/2 pi cubic centimeters
25 pi cubic centimeters
50/3 pi cubic centimeters
25/6 pi cubic centimeters
The volume of a cone is given by V_cone = (1/3) * pi * r^2 * h, where r is the radius and h is the height.
Given that the volume of the cone is (25/3) * pi, we can rewrite the formula as (25/3) * pi = (1/3) * pi * r^2 * h.
Since the height of the cone is equal to the sphere's diameter, we can write h = 2r, where r is the radius.
Substituting this into the equation, we have:
(25/3) * pi = (1/3) * pi * r^2 * 2r
Simplifying further,
25 = 2r^3
Dividing both sides by 2,
12.5 = r^3
Taking the cube root of both sides,
r = 2.5
Now we need to find the volume of the sphere, which is given by V_sphere = (4/3) * pi * r^3.
Substituting the radius r = 2.5 into the formula,
V_sphere = (4/3) * pi * (2.5)^3
V_sphere = (4/3) * pi * 15.625
V_sphere = 62.5/3 * pi
V_sphere = 20.833... * pi
So, the volume of the sphere is approximately 20.833... * pi cubic centimeters, which can be simplified to 20.833... pi cubic centimeters, or approximately 62.5 cubic centimeters.
Therefore, the correct answer is 62.5 cubic centimeters, or in terms of pi, 25/2 pi cubic centimeters. Answer: 25/2 pi cubic centimeters.