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

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A community garden in Brisbane features a compost bin designed as a composite solid: a rectangular prism base measuring 63 cm in length, 44 cm in width, and 44 cm in height, with a right circular cylinder of radius 22 cm and height 63 cm mounted on top. The cylinder is attached 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 ground 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 = 22 cmr = 22 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

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Worked solution

  1. The composite solid consists of a rectangular prism base and a right circular cylinder mounted on top. The base of the prism is on the ground and not included in the surface area. The interface between the prism and cylinder is not exposed, so it is not part of the external surface area.
  2. The surface area of the composite solid is the sum of: (1) the lateral surface area of the rectangular prism (excluding base and top), (2) the lateral surface area of the cylinder (excluding top and bottom), and (3) the top face of the prism (which is covered by the cylinder base, so not exposed), and (4) the top face of the cylinder (exposed).
  3. The lateral surface area of a rectangular prism is 2h(l + w), where l is length, w is width, and h is height. The base is not included, and the top is covered by the cylinder, so only the four vertical sides contribute.
  4. Substitute values: l = 63 cm, w = 44 cm, h = 44 cm. Lateral surface area = 2 × 44 × (63 + 44) = 88 × 107 = 9416 cm².
  5. The lateral surface area of a right circular cylinder is 2πrh, where r is radius and h is height. The base and top are not included in the external surface area because the base is attached to the prism and the top is exposed.
  6. Substitute values: r = 22 cm, h = 63 cm. Lateral surface area = 2 × π × 22 × 63 = 2772π cm².
  7. The top face of the cylinder is exposed and must be included. Area of a circle is πr². r = 22 cm, so area = π × 22² = 484π cm².
  8. The base of the cylinder is attached to the prism and not exposed. The top face of the prism is covered by the cylinder base, so it is not exposed. Therefore, no additional area from the prism top or cylinder base is included.
  9. Total surface area = lateral area of prism + lateral area of cylinder + top area of cylinder = 9416 + 2772π + 484π cm².
  10. Combine like terms: 2772π + 484π = 3256π cm².
  11. Total surface area = 9416 + 3256π cm². This is the exact expression in terms of π.
  12. The total surface area of the composite solid is 9416 + 3256π square centimetres, expressed as a single exact expression in terms of π.

Reference answer: 9416 + 3256π

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