Question

A right circular cylinder is inscribed in a sphere and the height of the cylinder is equal to the diameter of its bas Find the ratio of the volume of the sphere to that of the cylinder.

• A. 4:\(\sqrt{3} \)
• B. 4\(\sqrt{2} \):3
• C. 2\(\sqrt{2} \):1
• D. 1:2
• E. None of these

Answer 1

Answer: (E)

Step 1: Understand the problem.
We are given that a right circular cylinder is inscribed in a sphere, and the height of the cylinder is equal to the diameter of its base. We are asked to find the ratio of the volume of the sphere to the volume of the cylinder.

Step 2: Define the variables.
Let the radius of the sphere be \( R \). The cylinder is inscribed in the sphere, meaning that the height of the cylinder is equal to the diameter of the base of the cylinder. Let the radius of the cylinder’s base be \( r \), and its height be \( h \). Since the height of the cylinder is equal to the diameter of the base, we have \( h = 2r \).

Step 3: Find the volume of the sphere.
The volume \( V_{\text{sphere}} \) of a sphere is given by the formula:
\( V_{\text{sphere}} = \frac{4}{3} \pi R^3 \)

Step 4: Find the volume of the cylinder.
The volume \( V_{\text{cylinder}} \) of a cylinder is given by the formula:
\( V_{\text{cylinder}} = \pi r^2 h = \pi r^2 \times 2r = 2 \pi r^3 \)

Step 5: Relate the cylinder and the sphere.
Since the cylinder is inscribed in the sphere, the diagonal of the cylinder (which is the hypotenuse of the right triangle formed by the radius of the sphere, the radius of the cylinder, and half the height of the cylinder) must be equal to the diameter of the sphere. The diagonal \( d \) of the cylinder is given by the Pythagorean theorem:
\( d^2 = r^2 + \left( \frac{h}{2} \right)^2 \)
\( d^2 = r^2 + r^2 = 2r^2 \)
\( d = \sqrt{2}r \)
The diagonal of the cylinder is also equal to the diameter of the sphere, which is \( 2R \). Therefore:
\( \sqrt{2}r = 2R \)
\( r = \frac{2R}{\sqrt{2}} = \sqrt{2}R \)

Step 6: Find the ratio of the volumes.
Now that we know \( r = \sqrt{2}R \), we can substitute this into the formula for the volume of the cylinder:
\( V_{\text{cylinder}} = 2 \pi r^3 = 2 \pi (\sqrt{2}R)^3 = 2 \pi \times 2\sqrt{2} R^3 = 4\pi \sqrt{2} R^3 \)

Now, the ratio of the volume of the sphere to the volume of the cylinder is:
\( \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{4}{3} \pi R^3}{4\pi \sqrt{2} R^3} = \frac{4}{3} \times \frac{1}{4\sqrt{2}} = \frac{1}{3\sqrt{2}} \)

Step 7: Conclusion.
The ratio of the volume of the sphere to that of the cylinder is \( \frac{1}{3\sqrt{2}} \), which does not match any of the given options.

Final Answer:
The correct option is (E): None of these.