Question

The ratio between the volume of a sphere and volume of a circumscribing right circular cylinder is :

• A. 2 : 1
• B. 1 : 1
• C. 2 : 3
• D. 1 : 2

Hint:

This problem is a classic geometry question that highlights the relationship between a sphere and its circumscribing cylinder. The key is to correctly identify the dimensions of the cylinder in terms of the sphere's radius.

Answer 1

The Correct Option is C

Step 1: Understand the term 'circumscribing right circular cylinder'.
A circumscribing right circular cylinder for a sphere means that the cylinder just encloses the sphere such that the sphere touches the top, bottom, and curved surface of the cylinder.
For this to happen:
The radius of the cylinder ($R_{cyl}$) must be equal to the radius of the sphere ($R_{sph}$). So, $R_{cyl} = R_{sph} = R$.
The height of the cylinder ($H_{cyl}$) must be equal to the diameter of the sphere. So, $H_{cyl} = 2R_{sph} = 2R$. Step 2: Write down the formulas for the volume of a sphere and a cylinder.
Volume of a sphere ($V_{sph}$) = $\frac{4}{3}\pi R_{sph}^3$
Volume of a cylinder ($V_{cyl}$) = $\pi R_{cyl}^2 H_{cyl}$
Step 3: Substitute the relationships from Step 1 into the volume formulas.
For the sphere, let its radius be $R$. So, $V_{sph} = \frac{4}{3}\pi R^3$. For the circumscribing cylinder:
$R_{cyl} = R$
$H_{cyl} = 2R$
So, $V_{cyl} = \pi (R)^2 (2R) = 2\pi R^3$.
Step 4: Calculate the ratio of the volume of the sphere to the volume of the circumscribing cylinder.
Ratio = $\frac{V_{sph}}{V_{cyl}} = \frac{\frac{4}{3}\pi R^3}{2\pi R^3}$ Cancel out $\pi R^3$ from the numerator and denominator:
Ratio = $\frac{\frac{4}{3}}{2} = \frac{4}{3 \times 2} = \frac{4}{6} = \frac{2}{3}$ So, the ratio is 2 : 3.
Step 5: Compare with the given options.
The calculated ratio is 2 : 3, which matches option (3). (3) 2 : 3