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A composite solid is formed by attaching a right circular cylinder of radius 5 m and height 8 m to the top of a rectangular prism with base dimensions 10 m by 10 m and height 6 m. The cylinder is placed so that its base is perfectly aligned with the top face of the prism.
Calculate the exact total volume of the composite solid in cubic metres. Use the formula for the volume of a cylinder: pi * r^2 * h, and the formula for the volume of a rectangular prism: l * 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 and a right circular cylinder attached at their bases. The prism has base dimensions 10 m by 10 m and height 6 m. The cylinder has radius 5 m and height 8 m. The total volume is the sum of the individual volumes.
- Volume of a rectangular prism is given by V = l × w × h. Substitute l = 10 m, w = 10 m, h = 6 m.
- V_prism = 10 × 10 × 6 = 600 m³.
- Volume of a right circular cylinder is given by V = π × r² × h. Substitute r = 5 m, h = 8 m.
- V_cylinder = π × (5)² × 8 = π × 25 × 8 = 200π m³.
- Total volume = V_prism + V_cylinder = 600 + 200π m³.
- The expression 600 + 200π is already in exact form and combines both volumes.
- No further simplification is needed because the problem asks for a single exact expression in terms of π.
- Check: 200π is approximately 628.32, so total volume ≈ 600 + 628.32 = 1228.32 m³. This is reasonable for the given dimensions.
- The cylinder is fully attached and not overlapping; no volume is double-counted.
- The base of the cylinder (area = π×5² = 25π) is exactly equal to the top face of the prism (10×10 = 100 m²). Since 25π ≈ 78.54 < 100, the cylinder does not cover the entire top face, but the problem states it is placed so that its base is perfectly aligned, meaning it sits centrally and does not extend beyond. This does not affect volume calculation as no material is removed.
- The total volume of the composite solid is exactly 600 + 200π cubic metres.
Reference answer: 600 + 200pi
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