If the ratio of the diameters of two right circular cones of equal height be 3:4, then the ratio of their volume will be

Options

A. `3:4`
B. `9:16`
C. `16:9`
D. `27:64`

Answer

To find the ratio of the volumes of two right circular cones with equal heights and diameters in the ratio of 3:4, we can follow these steps:

### Step 1: Understand the relationship between diameter and radius
The diameter of a cone is related to its radius as follows:
- Diameter = 2 * Radius
Thus, if the diameters of the two cones are in the ratio 3:4, then their radii will be in the ratio:
- Radius of Cone 1 (r1) = 3k (for some constant k)
- Radius of Cone 2 (r2) = 4k

### Step 2: Write down the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 3: Set the height of both cones
Since the heights of both cones are equal, we can denote the height as \( h \).

### Step 4: Calculate the volumes of both cones
Now, we can calculate the volumes of both cones using their respective radii:
- Volume of Cone 1 (V1):
\[ V_1 = \frac{1}{3} \pi (r_1^2) h = \frac{1}{3} \pi (3k)^2 h = \frac{1}{3} \pi (9k^2) h = 3 \pi k^2 h \]

- Volume of Cone 2 (V2):
\[ V_2 = \frac{1}{3} \pi (r_2^2) h = \frac{1}{3} \pi (4k)^2 h = \frac{1}{3} \pi (16k^2) h = \frac{16}{3} \pi k^2 h \]

### Step 5: Find the ratio of the volumes
Now we can find the ratio of the volumes \( V_1 \) and \( V_2 \):
\[ \text{Ratio of volumes} = \frac{V_1}{V_2} = \frac{3 \pi k^2 h}{\frac{16}{3} \pi k^2 h} \]
The \( \pi \), \( k^2 \), and \( h \) cancel out:
\[ = \frac{3}{\frac{16}{3}} = \frac{3 \times 3}{16} = \frac{9}{16} \]

### Conclusion
Thus, the ratio of the volumes of the two cones is \( \frac{9}{16} \).

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