Let's solve the problem step by step. Given: - Q = 12 m³/s - S = .0015 - n = .015 - Depth (h) = 1.4 m - Bottom width (b) = 3 m - Side slope = 1.5:1 (horizontal:vertical) --- ## **Step 1: Área hidráulica (a)** **Trapezoidal Channel Area Formula:** \[ A = b \cdot h + z h^2 \] Where: - \(b\) = bottom width = 3 m - \(h\) = depth = 1.4 m - \(z\) = side slope (horizontal/vertical) = 1.5 **Calculate:** \[ A = 3 \cdot 1.4 + 1.5 \cdot (1.4)^2 \] \[ A = 4.2 + 1.5 \cdot 1.96 \] \[ A = 4.2 + 2.94 = 7.14~m^2 \] **Explanation:** The area is the bottom area plus the area from both side slopes. --- ## **Step 2: Perímetro mojado (b)** **Wetted Perimeter Formula:** \[ P = b + 2h\sqrt{1+z^2} \] **Calculate:** \[ P = 3 + 2 \cdot 1.4 \cdot \sqrt{1 + (1.5)^2} \] \[ P = 3 + 2.8 \cdot \sqrt{1 + 2.25} \] \[ P = 3 + 2.8 \cdot \sqrt{3.25} \] \[ \sqrt{3.25} \approx 1.803 \] \[ P = 3 + 2.8 \cdot 1.803 \] \[ P = 3 + 5.05 = 8.05~m \] **Explanation:** The wetted perimeter is the bottom plus the two side lengths in contact with water. --- ## **Final Answers (a & b):** **a) Área hidráulica:** \(7.14~m^2\) **b) Perímetro mojado:** \(8.05~m\) If you want to continue with the next parts, let me know!

Let's continue solving the remaining parts of the problem. --- ## **Step 3: Radio hidráulico (c)** **Hydraulic Radius Formula:** \[ R = \frac{A}{P} \] **Calculate:** Using the previously calculated values: - \(A = 7.14~m^2\) - \(P = 8.05~m\) \[ R = \frac{7.14}{8.05} \approx 0.885~m \] **Explanation:** The hydraulic radius is the area divided by the wetted perimeter, representing how effectively the channel can convey flow. --- ## **Step 4: Velocidad media del flujo (d)** **Velocity Formula using Manning's Equation:** \[ v = \frac{1}{n} R^{2/3} S^{1/2} \] **Calculate:** Using: - \(n = 0.015\) - \(R = 0.885~m\) - \(S = 0.0015\) \[ v = \frac{1}{0.015} (0.885)^{2/3} (0.0015)^{1/2} \] 1. Calculate \(R^{2/3}\): \[ R^{2/3} \approx 0.885^{0.6667} \approx 0.763 \] 2. Calculate \(S^{1/2}\): \[ S^{1/2} = \sqrt{0.0015} \approx 0.03873 \] 3. Combine: \[ v = \frac{1}{0.015} \cdot 0.763 \cdot 0.03873 \approx 1.960~m/s \] **Explanation:** The average flow velocity is derived from Manning's equation, which factors in roughness, hydraulic radius, and slope. --- ## **Step 5: Verificar caudal (e)** **Flow Rate Formula:** \[ Q = A \cdot v \] **Calculate:** Using: - \(A = 7.14~m^2\) - \(v \approx 1.960~m/s\) \[ Q = 7.14 \cdot 1.960 \approx 13.99~m^3/s \] **Check against design flow:** - **Given Design Flow:** \(Q_{design} = 12~m^3/s\) - **Calculated Flow:** \(Q \approx 13.99~m^3/s\) **Explanation:** We compare the calculated flow with the design flow. Since \(Q > Q_{design}\), the channel may be undersized for the flow conditions. --- ## **Final Answers (c, d, & e):** **c) Radio hidráulico:** \(0.885~m\) **d) Velocidad media del flujo:** \(1.960~m/s\) **e) Verificación del caudal:** \(Q \approx 13.99~m^3/s\) (exceeds design flow)