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 50 cm and height 53 cm to the top of a rectangular prism with base dimensions 100 cm by 100 cm and height 50 cm. The cylinder is placed so that its base is perfectly aligned with the top face of the prism, and the interface between the two solids 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 pi.

Circle measurementsradius = 50 cmr = 50 cm
Measurements from the question · not to scale.
Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

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 two solids is not exposed.
  2. Surface area of the rectangular prism (excluding base and interface): includes the lateral surface area of the prism and the top face (which is covered by the cylinder and thus not exposed).
  3. Lateral surface area of rectangular prism = 2 * (length + width) * height = 2 * (100 + 100) * 50 = 2 * 200 * 50 = 20,000 cm².
  4. The top face of the prism is covered by the cylinder's base, so it is not part of the external surface area.
  5. Surface area of the cylinder (excluding base and interface): includes the lateral surface area and the top circular face.
  6. Lateral surface area of cylinder = 2 * π * r * h = 2 * π * 50 * 53 = 5,300π cm².
  7. Top circular face of cylinder = π * r² = π * 50² = 2,500π cm².
  8. The base of the cylinder is attached to the prism and not exposed, so it is not included.
  9. Total surface area = lateral area of prism + lateral area of cylinder + top face of cylinder = 20,000 + 5,300π + 2,500π = 20,000 + 7,800π cm².
  10. The result is expressed as a single exact expression in terms of π.
  11. Check: 20,000 cm² (prism lateral) + 5,300π (cylinder lateral) + 2,500π (cylinder top) = 20,000 + 7,800π cm². All exposed surfaces accounted for.
  12. The total surface area of the composite solid is 20,000 + 7,800π square centimetres.

Reference answer: 20000 + 7800pi

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