1. **State the problem:** We need to find (i) the area of the quadrilateral site ABCD with given side lengths and angles, and (ii) the volume of soil excavated for a ditch 40 m long along one side with specified cross section. --- 2. **Analyze the shape and data:** - AB = 105 m (horizontal) - BC = 55 m, ∠B = 30° between AB and BC - CD = 70 m, ∠D = 80° between CD and AD - AD is vertical with a right angle at A We consider coordinate placements: - Place A at origin (0,0) - AB along the positive x-axis: B(105,0) - AD vertical, so D at (0,h) with unknown height $h$ --- 3. **Find coordinates of point C:** First, find coordinates of D: Since AD is vertical, $D = (0,h)$. Next, find point C using BC and CD: - Angle at B is 30° between AB (horizontal) and BC, so vector BC makes 30° above the x-axis. - Coordinate of C: $$ C = B + BC * (\cos30^\circ, \sin30^\circ) = (105 + 55 * \cos30^\circ, 0 + 55 * \sin30^\circ) $$ Calculate: - $\cos30^\circ = \sqrt{3}/2 \approx 0.866$ - $\sin30^\circ = 1/2 = 0.5$ Hence, $$ C_x = 105 + 55 * 0.866 = 105 + 47.63 = 152.63 $$ $$ C_y = 0 + 55 * 0.5 = 27.5 $$ So, $$ C \approx (152.63, 27.5) $$ --- 4. **Use distance CD = 70 m and angle at D = 80°: find $h$ (the height coordinate of D) and validate geometry:** Vector $\overrightarrow{CD} = D - C = (0 - 152.63, h - 27.5) = (-152.63, h - 27.5)$ Length of CD is 70: $$ \sqrt{(-152.63)^2 + (h - 27.5)^2} = 70 $$ Square both sides: $$ 152.63^2 + (h - 27.5)^2 = 4900 $$ $$ (h - 27.5)^2 = 4900 - 152.63^2 $$ Calculate $152.63^2$: $$ 152.63^2 = 23295.9 $$ So, $$ (h - 27.5)^2 = 4900 - 23295.9 = -18395.9 $$ Negative under square root means the assumed $h$ placement is inconsistent. Re-examine angle information. --- 5. **Check correct interpretation of angle D = 80° at vertex D between sides CD and AD:** Since AD is vertical, vector AD points straight up: $(0, h) - (0,0) = (0, h)$. Angle between vectors CD and AD is 80°. Use dot product formula to write equation for $h$: Vectors: $$ \overrightarrow{CD} = C - D = (152.63 - 0, 27.5 - h) = (152.63, 27.5 - h) $$ $$ \overrightarrow{AD} = A - D = (0 - 0, 0 - h) = (0, -h) $$ Angle between CD and AD is 80°: $$ \cos80^\circ = \frac{\overrightarrow{CD} \cdot \overrightarrow{AD}}{|CD||AD|} $$ Dot product: $$ \overrightarrow{CD} \cdot \overrightarrow{AD} = 152.63*0 + (27.5 - h)(-h) = -h(27.5 - h) = h(h - 27.5) $$ Length calculations: $$ |CD| = 70 $$ $$ |AD| = h $$ Rewrite: $$ \cos80^\circ = \frac{h(h - 27.5)}{70 * h} = \frac{h - 27.5}{70} $$ Calculate $\cos80^\circ \approx 0.1736$ Equation: $$ 0.1736 = \frac{h - 27.5}{70} $$ Solve for $h$: $$ h - 27.5 = 70 * 0.1736 = 12.152 $$ $$ h = 12.152 + 27.5 = 39.652 $$ --- 6. **Coordinates of D:** $$ D = (0, 39.652) $$ We have A(0,0), B(105,0), C(152.63, 27.5), D(0, 39.652) --- 7. **Calculate the area of quadrilateral ABCD using shoelace formula:** List points in order: $$ (x_1,y_1) = A(0,0) $$ $$ (x_2,y_2) = B(105,0) $$ $$ (x_3,y_3) = C(152.63, 27.5) $$ $$ (x_4,y_4) = D(0,39.652) $$ Shoelace formula area: $$ \text{Area} = \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1)| $$ Calculate terms: $$ S_1 = 0*0 + 105*27.5 + 152.63*39.652 + 0*0 = 0 + 2887.5 + 6046.73 + 0 = 8934.23 $$ $$ S_2 = 0*105 + 0*152.63 + 27.5*0 + 39.652*0 = 0 $$ Area: $$ = \frac{1}{2} |8934.23 - 0| = 4467.115 m^2 $$ --- 8. **Calculate volume of excavated soil:** Given cross-sectional trapezoid shape with base (along side) and height (depth) dimensions but no explicit cross section dimensions given. Assuming ditch along AB (length 40 m), and cross section corresponds to vertical dimension AD and ditch width. Since no ditch width or cross section dimensions are clearly stated except "40 m long ditch running along one side," the volume calculation depends on known cross-section area. Assuming cross-sectional area (cross-section dimensions) is width * depth, with width unknown. **User states cross-section dimensions are given but image data is missing; we assume cross-section area $A_{cs}$ is given but not specified here.** Because insufficient data for cross section, volume cannot be exactly computed. --- 9. **Conclusion:** (i) Area of development site ABCD is approximately $$\boxed{4467.1 \text{ m}^2}$$ (ii) Volume of the ditch excavation requires cross-sectional area; with length 40 m: $$ \text{Volume} = A_{cs} * 40 $$ Cross-sectional area must be provided to calculate volume. ---