NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Composite-object surface area and volume

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Required skills (click to reveal)

A composite solid is formed by attaching a right circular cylinder of radius 68 cm and height 74 cm to the top of a rectangular prism with base dimensions 136 cm by 136 cm and height 68 cm. The cylinder is placed so that its base is perfectly aligned with the top face of the prism, and the interface between them is not exposed.

The base of the prism rests on the floor and is not included in the surface area.

Calculate the exact total surface area of the composite solid in square centimetres, expressed as a single exact expression in terms of π.

Circle measurementsradius = 68 cmr = 68 cm
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Concept illustrations

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Subtract a rectangular corner cut-out. The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.10 cm6 cm2 cm3 cm54 cm²
The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.
Area and perimeter of a rectangle. 6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.6 units across4
6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The composite solid consists of a rectangular prism and a right circular cylinder attached at their bases. The base of the prism is not included in the surface area, and the interface between the prism and cylinder is not exposed.
  2. The surface area of the composite solid is the sum of: (1) the lateral surface area of the prism, (2) the lateral surface area of the cylinder, and (3) the top face of the prism (since the base is not included and the interface is hidden).
  3. The rectangular prism has base dimensions 136 cm × 136 cm and height 68 cm. The lateral surface area of a rectangular prism is 2 × (length + width) × height.
  4. Substitute values: 2 × (136 + 136) × 68 = 2 × 272 × 68. First, 2 × 272 = 544. Then, 544 × 68.
  5. Calculate 544 × 68: 544 × 60 = 32,640; 544 × 8 = 4,352. Add: 32,640 + 4,352 = 36,992 cm².
  6. The top face of the prism is a square with side 136 cm. Area = 136² = 18,496 cm².
  7. The cylinder has radius 68 cm and height 74 cm. The lateral surface area of a cylinder is 2πrh.
  8. Substitute values: 2 × π × 68 × 74. First, 2 × 68 = 136; 136 × 74.
  9. Calculate 136 × 74: 136 × 70 = 9,520; 136 × 4 = 544. Add: 9,520 + 544 = 10,064. So lateral surface area = 10,064π cm².
  10. The base of the cylinder is not exposed (attached to prism), and the top of the cylinder is exposed. The top face of the cylinder is a circle with area πr² = π × 68².
  11. Calculate 68² = 4,624. So top face area = 4,624π cm².
  12. Total surface area = lateral prism (36,992) + top face (18,496) + lateral cylinder (10,064π) + top of cylinder (4,624π). Combine like terms: 36,992 + 18,496 = 55,488 cm²; 10,064π + 4,624π = 14,688π cm². Final answer: 55,488 + 14,688π cm².

Reference answer: 55488 + 14688π

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