Question

A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside is Wπ. Express W as a common tion.

a. 4/3
b. 5/3
c. 2/3
d. 1/3

Answer 1

W = 2/3 because The ratio of the space inside the sphere and outside the inscribed cylinder is 2/3, as the difference in their volumes is 433.33π cubic units.

Step-by-step explanation:

Consider the right cylinder inscribed in the sphere. The diameter of the sphere is twice its radius, so the diameter is 10 units. Since the cylinder is inscribed, its height is equal to the diameter of the sphere, which is 10 units.

Now, the volume of the cylinder is given by V_cylinder = πr^2h, where r is the radius of the base and h is the height. Substituting r = 3 and h = 10, we get V_cylinder = 90π cubic units.

The volume of the sphere is given by V_sphere = (4/3)πr^3, where r is the radius. Substituting r = 5, we get V_sphere = 523.33π cubic units.

Now, the total volume inside the sphere and outside the cylinder is the difference between the volume of the sphere and the volume of the cylinder: V_total = V_sphere - V_cylinder = (523.33 - 90)π = 433.33π cubic units.

Finally, W is the ratio of the total volume to the volume of the cylinder: W = V_total / V_cylinder = (433.33π) / (90π) = 2/3. Therefore, the correct answer is W = 2/3.

Therefore, the correct answer is option C