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 school is ordering custom storage bins for science equipment. Each bin is a composite solid formed by attaching a right circular cylinder of radius 29 cm and height 93 cm to the top of a rectangular prism base with base dimensions 58 cm by 58 cm and height 29 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 one storage bin in square centimetres, excluding the base of the prism and the interface between the prism and cylinder. Express your answer as a single exact expression in terms of pi.

Circle measurementsradius = 29 cmr = 29 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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The storage bin is a composite solid made of a rectangular prism base and a right circular cylinder on top. The base of the prism is not included in the surface area, and the interface between the prism and cylinder is not exposed.
  2. The surface area to calculate includes: the lateral surface area of the rectangular prism, the lateral surface area of the cylinder, and the top face of the cylinder (since the base of the prism is excluded and the interface is not exposed).
  3. The rectangular prism has base dimensions 58 cm × 58 cm and height 29 cm. The lateral surface area of a rectangular prism is 2 × (length + width) × height.
  4. Substitute values: 2 × (58 + 58) × 29 = 2 × 116 × 29. First, 116 × 29 = (100 × 29) + (16 × 29) = 2900 + 464 = 3364. Then, 2 × 3364 = 6728 cm².
  5. The cylinder has radius 29 cm and height 93 cm. The lateral surface area of a cylinder is 2πrh.
  6. Substitute values: 2 × π × 29 × 93. First, 2 × 29 = 58. Then, 58 × 93 = (50 × 93) + (8 × 93) = 4650 + 744 = 5394. So, lateral surface area = 5394π cm².
  7. The top face of the cylinder is a circle with radius 29 cm. Its area is πr² = π × 29² = π × 841 cm².
  8. The base of the prism is not included, and the interface between the prism and cylinder is not exposed, so no areas are added or subtracted beyond the lateral and top surfaces.
  9. Total surface area = lateral area of prism + lateral area of cylinder + top area of cylinder = 6728 + 5394π + 841.
  10. Combine constant terms: 6728 + 841 = 7569. So, total surface area = 7569 + 5394π cm².
  11. The expression is already in exact form, as required. No further simplification is needed.
  12. The total surface area of one storage bin, excluding the base of the prism and the interface, is 7569 + 5394π cm².

Reference answer: 7569 + 5394π

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