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A composite solid is formed by attaching a right circular cylinder of radius 3 m and height 5 m to the top of a rectangular prism with base dimensions 6 m by 6 m and height 4 m. The cylinder is placed 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. (Note: The base of the prism is on the ground and not exposed; the top of the cylinder is exposed, and the interface between the cylinder and prism is not included in the surface area.)
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Worked solution
- The composite solid consists of a rectangular prism (base 6 m × 6 m, height 4 m) and a right circular cylinder (radius 3 m, height 5 m) attached on top. The base of the prism is on the ground and not exposed. The interface between the prism and cylinder is not included in the surface area. We must calculate the total external surface area.
- Surface area of the rectangular prism (excluding base and top face where cylinder is attached): lateral surface area + one base (bottom) + one top face (exposed). The base is on the ground and not exposed, so only the bottom base is exposed. The top face is covered by the cylinder, so it is not exposed.
- Lateral surface area of rectangular prism = 2 × (length + width) × height = 2 × (6 + 6) × 4 = 2 × 12 × 4 = 96 m².
- Bottom base area = length × width = 6 × 6 = 36 m².
- Top face of prism is covered by cylinder, so not included. Therefore, prism contributes 96 + 36 = 132 m².
- Cylinder surface area (excluding base, since it is attached to prism): lateral surface area + top face (exposed).
- Lateral surface area of cylinder = 2 × π × radius × height = 2 × π × 3 × 5 = 30π m².
- Top face area of cylinder = π × radius² = π × 3² = 9π m².
- Total cylinder surface area (exposed) = 30π + 9π = 39π m².
- Total surface area = prism contribution + cylinder contribution = 132 + 39π m².
- The interface between prism and cylinder is not exposed, so no area is double-counted or subtracted. The base of the prism is not exposed, so only the bottom base is included.
- Final answer is 132 + 39π square metres, as the surface area includes all external exposed faces.
Reference answer: 132 + 39pi
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