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A composite solid is formed by placing a right circular cylinder of radius 4 m and height 7 m on top of a rectangular prism with base dimensions 8 m by 8 m and height 5 m. The cylinder is attached so that its base is perfectly aligned with the top face of the prism.
Calculate the exact total surface area of the composite solid in square metres, excluding the interface between the cylinder and prism and excluding the base of the prism (which is on the ground). Use the formula for the surface area of a cylinder (2 * pi * r * h + 2 * pi * r^2) and the surface area of a rectangular prism (2 * (l * w + l * h + w * h)).
Express your answer as a single exact expression in terms of pi.
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Worked solution
- The composite solid consists of a rectangular prism (base 8 m × 8 m, height 5 m) and a right circular cylinder (radius 4 m, height 7 m) placed on top. The base of the prism is on the ground, and the interface between the cylinder and prism is not included in the surface area.
- We must calculate the total external surface area, excluding the base of the prism (on ground) and the interface between the cylinder and prism.
- Surface area of the rectangular prism (excluding base and top interface): the prism has 5 exposed faces: 4 lateral faces and the base is excluded. The top face is covered by the cylinder, so it is not exposed.
- Lateral surface area of the prism: 4 × (length × height) = 4 × (8 m × 5 m) = 4 × 40 m² = 160 m².
- The base of the prism is on the ground and excluded, so we do not include the 8 m × 8 m base area.
- The top face of the prism (8 m × 8 m) is covered by the cylinder, so it is not exposed and excluded.
- Surface area of the cylinder (excluding base and top): the base is attached to the prism and not exposed, and the top is exposed. So we include only the lateral surface area and the top circular face.
- Lateral surface area of cylinder: 2 × π × r × h = 2 × π × 4 m × 7 m = 56π m².
- Top circular face of cylinder: π × r² = π × (4 m)² = 16π m².
- Total surface area = lateral prism area + lateral cylinder area + top cylinder face = 160 m² + 56π m² + 16π m² = 160 + 72π m².
- The interface between cylinder and prism is not exposed, and the base of the prism is on the ground, so both are excluded as required.
- The total surface area is 160 + 72π square metres, which is the exact expression required.
Reference answer: 160 + 72pi
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