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 library storage unit in Melbourne is designed as a composite solid consisting of a rectangular prism base with dimensions 34 cm in length, 17 cm in width, and 51 cm in height, topped by a right circular cylinder of radius 17 cm and height 34 cm, placed so that its base is perfectly aligned with the top face of the prism. The cylinder is attached such that the base of the cylinder is not exposed.

Calculate the exact total surface area of the composite solid in square centimetres, excluding the base of the prism (which rests on the floor) and the interface between the prism and cylinder. Express your answer as a single exact expression in terms of pi.

Circle measurementsradius = 17 cmr = 17 cm
Measurements from the question · not to scale.
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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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The composite solid consists of a rectangular prism base (34 cm × 17 cm × 51 cm) and a right circular cylinder (radius 17 cm, height 34 cm) placed on top. The base of the prism rests on the floor and is not included in the surface area. The interface between the prism and cylinder is not exposed and must be excluded.
  2. The total surface area is the sum of: (1) the lateral surface area of the prism, (2) the top face of the prism (exposed), (3) the lateral surface area of the cylinder, and (4) the top face of the cylinder. The base of the prism and the interface are excluded.
  3. Lateral surface area of a rectangular prism = 2 × (length × height + width × height). Substituting: 2 × (34 × 51 + 17 × 51) = 2 × (1734 + 867) = 2 × 2601 = 5202 cm².
  4. Top face of the prism is a rectangle: length × width = 34 × 17 = 578 cm².
  5. Lateral surface area of a right circular cylinder = 2 × π × radius × height. Substituting: 2 × π × 17 × 34 = 2 × π × 578 = 1156π cm².
  6. Top face of the cylinder is a circle: π × radius² = π × 17² = π × 289 = 289π cm².
  7. Total surface area = lateral prism + top prism + lateral cylinder + top cylinder = 5202 + 578 + 1156π + 289π.
  8. Combine like terms: 5202 + 578 = 5780 cm², and 1156π + 289π = 1445π cm².
  9. The total surface area is therefore 5780 + 1445π cm².
  10. All values are exact. No rounding is required. The final answer is expressed as a single exact expression in terms of π.
  11. The surface area excludes the base of the prism and the interface between prism and cylinder, as required. All components are accounted for.
  12. The total surface area of the composite solid is 5780 + 1445π square centimetres.

Reference answer: 5780 + 1445π

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