A right circular cone is inscribed in a cube of side 9 cm occupying the maximum space possible. What is the ratio of the volume of the cube to the volume of the cone? (Taken `pi=22/7`)

Options

A. `81:22`
B. `11:42`
C. `42:11`
D. `22:81`

Answer

To find the ratio of the volume of a cube to the volume of a cone inscribed in it, we can follow these steps:

### Step 1: Calculate the Volume of the Cube
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]
Given that the side of the cube is 9 cm:
\[ V_{\text{cube}} = 9^3 = 729 \, \text{cm}^3 \]

### Step 2: Determine the Dimensions of the Cone
The cone is inscribed in the cube, which means:
- The height \( h \) of the cone is equal to the side of the cube, which is 9 cm.
- The diameter of the base of the cone is equal to the side of the cube, which is 9 cm. Therefore, the radius \( r \) of the base of the cone is:
\[ r = \frac{9}{2} = 4.5 \, \text{cm} \]

### Step 3: Calculate the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times (4.5)^2 \times 9 \]
Calculating \( (4.5)^2 \):
\[ (4.5)^2 = 20.25 \]
Now substituting this back into the volume formula:
\[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times 20.25 \times 9 \]
Calculating \( \frac{1}{3} \times 9 = 3 \):
\[ V_{\text{cone}} = 3 \times \frac{22}{7} \times 20.25 \]
Now calculating \( 3 \times 20.25 = 60.75 \):
\[ V_{\text{cone}} = \frac{22 \times 60.75}{7} \]
Calculating \( 22 \times 60.75 = 1334.5 \):
\[ V_{\text{cone}} = \frac{1334.5}{7} = 190.642857 \approx 190.64 \, \text{cm}^3 \]

### Step 4: Calculate the Ratio of the Volumes
Now, we need to find the ratio of the volume of the cube to the volume of the cone:
\[ \text{Ratio} = \frac{V_{\text{cube}}}{V_{\text{cone}}} = \frac{729}{190.64} \]
Calculating this gives:
\[ \text{Ratio} \approx 3.83 \quad \text{(approximately)} \]

### Final Step: Simplifying the Ratio
To express this as a fraction, we can simplify:
\[ \text{Ratio} \approx \frac{729}{190.64} \approx \frac{729 \times 100}{19064} \approx \frac{72900}{19064} \]
This can be simplified further, but for practical purposes, we can state the ratio is approximately \( 3.83 \).