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A tube A of radius r is connected to two...

A tube A of radius r is connected to two other tubes B and C with the help of a junction valve. The tubes B and C have radii `(r )/(2)` and `(r )/(3)` and the flow velocity in each is v and 3v resopectively. If the flow velocity in tube A is `(n)/(m)v`, when n and m are integers, then what is the value of `n+m` ?

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To solve the problem, we will use the principle of conservation of mass, which is expressed through the equation of continuity for fluid flow. The equation states that the mass flow rate must be constant throughout the system. ### Step-by-Step Solution: 1. **Identify the Areas of the Tubes**: - The area \( A \) of tube A with radius \( r \) is given by: \[ A = \pi r^2 \] - The area \( A_B \) of tube B with radius \( \frac{r}{2} \) is: \[ A_B = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \] - The area \( A_C \) of tube C with radius \( \frac{r}{3} \) is: \[ A_C = \pi \left(\frac{r}{3}\right)^2 = \pi \frac{r^2}{9} \] 2. **Write the Flow Velocities**: - The flow velocity in tube B is \( v \). - The flow velocity in tube C is \( 3v \). - Let the flow velocity in tube A be \( v_A = \frac{n}{m}v \). 3. **Apply the Equation of Continuity**: The equation of continuity states that the total inflow must equal the total outflow: \[ A \cdot v_A = A_B \cdot v + A_C \cdot 3v \] Substituting the areas and velocities: \[ \pi r^2 \cdot \frac{n}{m}v = \left(\pi \frac{r^2}{4}\right) \cdot v + \left(\pi \frac{r^2}{9}\right) \cdot 3v \] 4. **Simplify the Equation**: Cancel \( \pi r^2 \) from both sides: \[ \frac{n}{m}v = \frac{1}{4}v + \frac{3}{9}v \] Simplifying the right side: \[ \frac{n}{m} = \frac{1}{4} + \frac{1}{3} \] 5. **Find a Common Denominator**: The common denominator for 4 and 3 is 12: \[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{3} = \frac{4}{12} \] Therefore: \[ \frac{n}{m} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \] 6. **Identify Values of n and m**: Here, \( n = 7 \) and \( m = 12 \). 7. **Calculate n + m**: \[ n + m = 7 + 12 = 19 \] ### Final Answer: The value of \( n + m \) is \( 19 \).
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