Water flows through a non-uniform tube of area of cross section A, B and C whose values are 25, 15 and `35cm^(2)` resprectively. The ratio of the velocities of water at the sections A,B and C is

Options

A. `5:3:7`
B. `7:3:5`
C. `21:35:15`
D. `1:1:1`

Explanation

To find the ratio of the velocities of water at sections A, B, and C in a non-uniform tube, we can use the principle of conservation of mass, which states that the mass flow rate must remain constant throughout the tube. This means that the product of the cross-sectional area (A) and the velocity (V) at any point in the tube is constant.

### Step-by-step solution:

1. **Identify the Areas**:
- Area at section A, \( A_A = 25 \, \text{cm}^2 \)
- Area at section B, \( A_B = 15 \, \text{cm}^2 \)
- Area at section C, \( A_C = 35 \, \text{cm}^2 \)

2. **Apply the Continuity Equation**:
The continuity equation states that:
\[ A_A \cdot V_A = A_B \cdot V_B = A_C \cdot V_C \]
where \( V_A, V_B, \) and \( V_C \) are the velocities at sections A, B, and C, respectively.

3. **Set Up the Equations**:
From the continuity equation, we can write:
\[ 25 \cdot V_A = 15 \cdot V_B = 35 \cdot V_C \]

4. **Express Velocities in Terms of Each Other**:
We can express the velocities in terms of a common variable:
\[ V_A = \frac{15}{25} V_B \quad \text{and} \quad V_A = \frac{35}{25} V_C \]
Simplifying these gives:
\[ V_A = \frac{3}{5} V_B \quad \text{and} \quad V_A = \frac{7}{5} V_C \]

5. **Find Ratios**:
Now we can find the ratios of the velocities:
- From \( V_A = \frac{3}{5} V_B \), we can express \( V_B \) in terms of \( V_A \):
\[ V_B = \frac{5}{3} V_A \]
- From \( V_A = \frac{7}{5} V_C \), we can express \( V_C \) in terms of \( V_A \):
\[ V_C = \frac{5}{7} V_A \]

6. **Combine the Ratios**:
Now we can write the ratios of the velocities:
\[ V_A : V_B : V_C = V_A : \frac{5}{3} V_A : \frac{5}{7} V_A \]
This simplifies to:
\[ 1 : \frac{5}{3} : \frac{5}{7} \]

7. **Clear the Fractions**:
To eliminate the fractions, we can multiply through by the least common multiple of the denominators (3 and 7):
\[ 1 \cdot 21 : \frac{5}{3} \cdot 21 : \frac{5}{7} \cdot 21 = 21 : 35 : 15 \]

8. **Final Ratio**:
Thus, the final ratio of the velocities is:
\[ V_A : V_B : V_C = 21 : 35 : 15 \]

### Final Answer:
The ratio of the velocities of water at sections A, B, and C is \( 21 : 35 : 15 \).