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 15 cm and height 78 cm to the top of a rectangular prism with base dimensions 30 cm by 30 cm and height 15 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 = 15 cmr = 15 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 the lateral surface areas of both solids, plus the area of the base of the cylinder (since the base of the prism is not exposed and the interface is hidden).
  3. Calculate the lateral surface area of the rectangular prism: 2 × (length × height + width × height). The base is 30 cm × 30 cm, and the height is 15 cm.
  4. Lateral surface area of prism = 2 × (30 × 15 + 30 × 15) = 2 × (450 + 450) = 2 × 900 = 1800 cm².
  5. Calculate the lateral surface area of the cylinder: 2πrh. Radius r = 15 cm, height h = 78 cm.
  6. Lateral surface area of cylinder = 2 × π × 15 × 78 = 2 × 15 × 78 × π = 2340π cm².
  7. Calculate the base area of the cylinder (which is exposed): πr². r = 15 cm.
  8. Base area = π × 15² = π × 225 = 225π cm².
  9. Total surface area = lateral area of prism + lateral area of cylinder + base area of cylinder.
  10. Total surface area = 1800 + 2340π + 225π = 1800 + (2340 + 225)π = 1800 + 2565π cm².
  11. The interface between the cylinder and prism is not exposed, so no area is subtracted. The base of the prism is not included, so no base area is added.
  12. The total surface area is 1800 + 2565π cm², expressed as a single exact expression in terms of pi.

Reference answer: 1800 + 2565π cm²

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