The ratio of diameter and higher of a right circular cylinder is 4 : 7. The ratio of heights of cylinders is 5 : 2. What is the ratio of the volume of cylinder?

Options

A. `40:49`
B. `8:7`
C. `9:7`
D. `17:7`

Explanation

To find the ratio of the volumes of two right circular cylinders given the ratios of their diameters and heights, we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Express Diameter in Terms of Radius
The diameter \( d \) of a cylinder is related to the radius \( r \) by the formula:
\[ d = 2r \quad \Rightarrow \quad r = \frac{d}{2} \]

### Step 3: Set Up the Ratios
Given the ratio of the diameters of the two cylinders is \( 4:7 \), we can express this as:
\[ \frac{d_1}{d_2} = \frac{4}{7} \]
From this, we can find the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{d_1/2}{d_2/2} = \frac{d_1}{d_2} = \frac{4}{7} \]

### Step 4: Use the Height Ratios
We are also given the ratio of the heights of the cylinders as \( 5:2 \):
\[ \frac{h_1}{h_2} = \frac{5}{2} \]

### Step 5: Write the Volume Ratio
Now, substituting the ratios into the volume formula:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \]
This simplifies to:
\[ \frac{V_1}{V_2} = \frac{r_1^2}{r_2^2} \cdot \frac{h_1}{h_2} \]

### Step 6: Substitute the Ratios
Substituting the ratios we found:
\[ \frac{V_1}{V_2} = \left(\frac{4}{7}\right)^2 \cdot \frac{5}{2} \]
Calculating this gives:
\[ \frac{V_1}{V_2} = \frac{16}{49} \cdot \frac{5}{2} = \frac{16 \cdot 5}{49 \cdot 2} = \frac{80}{98} = \frac{40}{49} \]

### Conclusion
Thus, the ratio of the volumes of the two cylinders is:
\[ \frac{V_1}{V_2} = \frac{40}{49} \]