# Step-by-Step Solution Let's break it down using the **continuity equation** and **Bernoulli's equation**, as guided by the hints. --- ## **Given Data** - **At A:** - Velocity, \( v_A = 15 \) ft/s - Pressure, \( P_A = 30 \) psi - Diameter at A, \( d_A = 6 \) in \( = 0.5 \) ft - **At B:** - Height above A: \( h = 12 \) ft - Diameter at B, \( d_B = 3 \) in \( = 0.25 \) ft --- ## **1. Find the Velocity at B (Continuity Equation)** The **continuity equation** for incompressible fluids states: \[ A_A v_A = A_B v_B \] Where \( A \) is the cross-sectional area. ### **Calculate Areas:** - \( A_A = \pi (d_A/2)^2 = \pi (0.25)^2 = 0.196 \) ft² - \( A_B = \pi (d_B/2)^2 = \pi (0.125)^2 = 0.0491 \) ft² ### **Apply Continuity:** \[ A_A v_A = A_B v_B \] \[ 0.196 \times 15 = 0.0491 \times v_B \] \[ 2.94 = 0.0491 \times v_B \] \[ v_B = \frac{2.94}{0.0491} \approx 59.9 \text{ ft/s} \] --- ## **2. Find the Pressure at B (Bernoulli's Equation)** The **Bernoulli equation** (ignoring friction) between A and B: \[ \frac{P_A}{\gamma} + z_A + \frac{v_A^2}{2g} = \frac{P_B}{\gamma} + z_B + \frac{v_B^2}{2g} \] - \( \gamma \): Specific weight of water (\( 62.4 \) lb/ft³) - \( g \): Acceleration due to gravity (\( 32.2 \) ft/s²) - Heights: \( z_B = z_A + 12 \) ft ### **Plug in values:** Let’s set \( z_A = 0 \), so \( z_B = 12 \) ft. \[ \frac{P_A}{62.4} + 0 + \frac{15^2}{2 \times 32.2} = \frac{P_B}{62.4} + 12 + \frac{59.9^2}{2 \times 32.2} \] #### **Calculate each term:** - \( \frac{P_A}{62.4} = \frac{30 \times 144}{62.4} = \frac{4320}{62.4} = 69.23 \) ft (since 1 psi = 144 lb/ft²) - \( \frac{15^2}{2 \times 32.2} = \frac{225}{64.4} = 3.50 \) ft - \( \frac{59.9^2}{2 \times 32.2} = \frac{3588}{64.4} = 55.73 \) ft #### **Set up equation:** \[ 69.23 + 0 + 3.50 = \frac{P_B}{62.4} + 12 + 55.73 \] \[ 72.73 = \frac{P_B}{62.4} + 67.73 \] \[ 72.73 - 67.73 = \frac{P_B}{62.4} \] \[ 5.00 = \frac{P_B}{62.4} \] \[ P_B = 5.00 \times 62.4 = 312 \text{ lb/ft}^2 \] Convert to psi: \[ P_B = \frac{312}{144} = 2.17 \text{ psi} \] --- ## **Final Answers** ### **1. Velocity at B** \[ \boxed{59.9\ \text{ft/s}} \] ### **2. Pressure at B** \[ \boxed{2.17\ \text{psi}} \] --- ## **Summary Table** | Location | Velocity (ft/s) | Pressure (psi) | |----------|-----------------|---------------| | A | 15 | 30 | | B | 59.9 | 2.17 | --- **Let me know if you want to see the step-by-step math in more detail or have questions about the equations!**