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 16 cm and height 69 cm to the top of a rectangular prism with base dimensions 32 cm by 32 cm and height 16 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 = 16 cmr = 16 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 two solids 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. Calculate the lateral surface area of the rectangular prism. The prism has base dimensions 32 cm × 32 cm and height 16 cm. The lateral surface area is the perimeter of the base times the height: 2 × (32 + 32) × 16.
  4. Perimeter of base = 2 × (32 + 32) = 2 × 64 = 128 cm. Lateral surface area = 128 × 16 = 2048 cm².
  5. The top face of the prism is a square of side 32 cm. Area = 32 × 32 = 1024 cm².
  6. Calculate the lateral surface area of the cylinder. The cylinder has radius 16 cm and height 69 cm. Lateral surface area = 2 × π × r × h = 2 × π × 16 × 69.
  7. 2 × 16 × 69 = 32 × 69 = 2208. So lateral surface area = 2208π cm².
  8. The base of the cylinder is attached to the prism and not exposed. The top of the cylinder is exposed and must be included in the surface area.
  9. Area of the top circular face of the cylinder = π × r² = π × 16² = 256π cm².
  10. Total surface area = lateral area of prism + top face of prism + lateral area of cylinder + top face of cylinder.
  11. Total surface area = 2048 + 1024 + 2208π + 256π = (2048 + 1024) + (2208 + 256)π = 3072 + 2464π cm².
  12. The total surface area is 3072 + 2464π square centimetres, expressed as a single exact expression in terms of pi.

Reference answer: 3072 + 2464π cm²

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