1. **Problem statement:** Find the total surface area of a right pyramid VABCD with a rectangular base of length 16 cm and width 14 cm, and slant edge length 25 cm. 2. **Formula and important rules:** - Total surface area (TSA) of a pyramid = Base area + Lateral surface area. - Base area for rectangle = length \times width. - Lateral surface area = sum of areas of triangular faces. - Each triangular face area = \frac{1}{2} \times base \times slant height. - Slant height here is the height of each triangular face, which can be found using Pythagoras theorem if needed. 3. **Calculate base area:** $$\text{Base area} = 16 \times 14 = 224 \text{ cm}^2$$ 4. **Find the height of the pyramid:** - The slant edge is 25 cm, which is the length from the apex V to a vertex of the base. - Half the base diagonal is needed to find the vertical height. - Base diagonal $$d = \sqrt{16^2 + 14^2} = \sqrt{256 + 196} = \sqrt{452} \approx 21.26 \text{ cm}$$ - Half diagonal $$= \frac{21.26}{2} = 10.63 \text{ cm}$$ - Using Pythagoras theorem for height $$h$$: $$h = \sqrt{25^2 - 10.63^2} = \sqrt{625 - 113.04} = \sqrt{511.96} \approx 22.63 \text{ cm}$$ 5. **Find slant heights of triangular faces:** - The pyramid has two pairs of triangular faces with different base lengths (16 cm and 14 cm). - Slant height for faces with base 16 cm is the height of the triangle with base 16 cm and slant edge 25 cm. - The slant height for these faces is the height of the triangular face, which is the distance from apex to midpoint of base edge. 6. **Calculate slant heights for each face:** - For length 16 cm side: Half base = 8 cm Slant height $$l_1 = \sqrt{25^2 - 8^2} = \sqrt{625 - 64} = \sqrt{561} \approx 23.7 \text{ cm}$$ - For width 14 cm side: Half base = 7 cm Slant height $$l_2 = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \text{ cm}$$ 7. **Calculate lateral surface area:** - Two triangles with base 16 cm and height 23.7 cm: $$2 \times \frac{1}{2} \times 16 \times 23.7 = 2 \times 8 \times 23.7 = 379.2 \text{ cm}^2$$ - Two triangles with base 14 cm and height 24 cm: $$2 \times \frac{1}{2} \times 14 \times 24 = 2 \times 7 \times 24 = 336 \text{ cm}^2$$ - Total lateral area = $$379.2 + 336 = 715.2 \text{ cm}^2$$ 8. **Calculate total surface area:** $$\text{TSA} = \text{Base area} + \text{Lateral area} = 224 + 715.2 = 939.2 \text{ cm}^2$$ 9. **Round to 3 significant figures:** $$\boxed{939 \text{ cm}^2}$$