1. **State the problem:** Calculate the hydrostatic force on the dam face when the water level is 52 meters below the top of the dam. The dam face is shaped like an isosceles trapezoid with a lower base of 74 m, an upper base of 126 m, and a height of 144 m. The weight density of water is 9800 N/m³. 2. **Understand the shape and water depth:** The water level is 52 m below the top, so the depth of water against the dam is $144 - 52 = 92$ meters. 3. **Calculate the width of the dam face at the water level:** The trapezoid's width changes linearly from 126 m at the top to 74 m at the bottom over 144 m height. The width at depth $y$ meters from the bottom is given by linear interpolation: $$w(y) = 74 + \frac{126 - 74}{144} y = 74 + \frac{52}{144} y = 74 + \frac{13}{36} y$$ At water depth $92$ m from the bottom: $$w(92) = 74 + \frac{13}{36} \times 92 = 74 + \frac{1196}{36} = 74 + 33.22 = 107.22 \text{ meters}$$ 4. **Calculate the area of the submerged trapezoid face:** The submerged height is 92 m, the width at the bottom is 74 m, and at the water level is 107.22 m. Area $A$ of trapezoid: $$A = \frac{(b_1 + b_2)}{2} \times h = \frac{74 + 107.22}{2} \times 92 = 90.61 \times 92 = 8336.92 \text{ m}^2$$ 5. **Calculate the hydrostatic force:** Hydrostatic force $F$ is given by: $$F = \rho g A \bar{h}$$ where $\rho g = 9800$ N/m³ is the weight density of water, $A$ is the area, and $\bar{h}$ is the depth of the centroid of the submerged area from the water surface. 6. **Find the centroid depth $\bar{h}$:** The centroid of a trapezoid from the bottom is: $$\bar{y} = \frac{h}{3} \times \frac{2b_1 + b_2}{b_1 + b_2} = \frac{92}{3} \times \frac{2 \times 74 + 107.22}{74 + 107.22} = 30.67 \times \frac{255.22}{181.22} = 30.67 \times 1.408 = 43.17 \text{ m}$$ The depth of centroid from the water surface is: $$\bar{h} = 92 - 43.17 = 48.83 \text{ m}$$ 7. **Calculate the hydrostatic force:** $$F = 9800 \times 8336.92 \times 48.83 = 3,993,000,000 \text{ N}$$ **Final answer:** The hydrostatic force on the dam face is approximately $3.99 \times 10^9$ Newtons.