An ideal fluid is flowing in a pipe in streamline flow. Pipe has maximum and minimum diameter of 6.4 cm and 4.8 cm respectively. Find out the ratio of minimum to maximum velocity.

Options

A. `(81)/(256)`
B. `(sqrt3)/(2)`
C. `(3)/(4)`
D. `(9)/(16)`

Answer

To find the ratio of minimum to maximum velocity of an ideal fluid flowing through a pipe with varying diameters, we can use the principle of conservation of mass, which states that the mass flow rate must remain constant throughout the pipe. This leads us to the equation of continuity for fluids.

### Step-by-Step Solution:

1. **Identify the diameters**:
- Maximum diameter (d1) = 6.4 cm
- Minimum diameter (d2) = 4.8 cm

2. **Calculate the cross-sectional areas**:
The cross-sectional area (A) of a pipe can be calculated using the formula:
\[ A = \frac{\pi d^2}{4} \]
- For maximum diameter (d1 = 6.4 cm):
\[ A_1 = \frac{\pi (6.4)^2}{4} = \frac{\pi \times 40.96}{4} = 10.24\pi \, \text{cm}^2 \]
- For minimum diameter (d2 = 4.8 cm):
\[ A_2 = \frac{\pi (4.8)^2}{4} = \frac{\pi \times 23.04}{4} = 5.76\pi \, \text{cm}^2 \]

3. **Use the equation of continuity**:
According to the equation of continuity for incompressible fluids:
\[ A_1 v_1 = A_2 v_2 \]
where \(v_1\) is the velocity at the maximum diameter and \(v_2\) is the velocity at the minimum diameter.

4. **Express velocities in terms of areas**:
Rearranging the equation gives:
\[ v_2 = \frac{A_1}{A_2} v_1 \]

5. **Substituting areas**:
Substitute the areas calculated:
\[ v_2 = \frac{10.24\pi}{5.76\pi} v_1 \]
The \(\pi\) cancels out:
\[ v_2 = \frac{10.24}{5.76} v_1 \]

6. **Calculate the ratio of velocities**:
The ratio of minimum to maximum velocity is:
\[ \frac{v_2}{v_1} = \frac{10.24}{5.76} \]

7. **Simplifying the fraction**:
To simplify:
\[ \frac{10.24}{5.76} = \frac{10.24 \div 5.76}{5.76 \div 5.76} = \frac{1.7778}{1} \approx \frac{9}{16} \]

8. **Final ratio**:
Thus, the ratio of minimum to maximum velocity is:
\[ \frac{v_{\text{min}}}{v_{\text{max}}} = \frac{9}{16} \]

### Conclusion:
The ratio of minimum to maximum velocity is \( \frac{9}{16} \).