1. **Stating the problem:** We need to find the cross-sectional area of the entrance area based on the given dimensions and shape. 2. **Understanding the shape:** The shape is star-like with multiple polygonal sections. We will break it down into simpler geometric shapes (triangles, rectangles, trapezoids) to calculate the total area. 3. **Given dimensions:** - Heights and lengths: 3.51 m, 2.37 m, 3.34 m, 6.8 m, 1.47 m, 4.5 m, 1.35 m, 23 m, 7.3 m - Angle: 65° 4. **Approach:** - Identify each polygonal section and calculate its area separately. - Use formulas for triangles: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ - Use formulas for rectangles: $$\text{Area} = \text{length} \times \text{width}$$ - Use formulas for trapezoids: $$\text{Area} = \frac{1}{2} (a + b) h$$ where $a$ and $b$ are parallel sides and $h$ is height. 5. **Calculations:** - Calculate the area of the top triangle using height 3.51 m and base calculated from 7.3 m and angle 65°: $$\text{base} = 2 \times 7.3 \times \cos(65^\circ) = 2 \times 7.3 \times 0.4226 = 6.17\,m$$ $$\text{Area}_{top} = \frac{1}{2} \times 6.17 \times 3.51 = 10.82\,m^2$$ - Calculate the area of the left polygon (approximate as rectangle + triangle): Rectangle: length 4.5 m, width 1.47 m $$\text{Area}_{rect} = 4.5 \times 1.47 = 6.615\,m^2$$ Triangle: base 1.35 m, height 2.37 m $$\text{Area}_{tri} = \frac{1}{2} \times 1.35 \times 2.37 = 1.60\,m^2$$ Total left polygon area: $$6.615 + 1.60 = 8.215\,m^2$$ - Calculate the area of the right polygon (approximate as rectangle + triangle): Rectangle: length 3.34 m, width 1.47 m $$\text{Area}_{rect} = 3.34 \times 1.47 = 4.91\,m^2$$ Triangle: base 1.35 m, height 2.37 m $$\text{Area}_{tri} = 1.60\,m^2$$ (same as left) Total right polygon area: $$4.91 + 1.60 = 6.51\,m^2$$ - Calculate the central lower trapezoid area with parallel sides 6.8 m and 23 m, height 1.35 m: $$\text{Area}_{trap} = \frac{1}{2} (6.8 + 23) \times 1.35 = \frac{1}{2} \times 29.8 \times 1.35 = 20.07\,m^2$$ 6. **Sum all areas:** $$\text{Total area} = 10.82 + 8.215 + 6.51 + 20.07 = 45.615\,m^2$$ 7. **Final answer:** The cross-sectional area of the entrance is approximately **45.62 m²**.