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 school is ordering custom storage bins for science equipment. Each bin is a composite solid made by attaching a right circular cylinder of radius 41 cm and height 79 cm to the top of a rectangular prism base with base dimensions 82 cm by 82 cm and height 41 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. Calculate the exact total surface area of one storage bin 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 = 41 cmr = 41 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 rests on the floor, so its bottom face is not part of the external surface area. The interface between the prism and cylinder is not exposed, so that area is excluded.
  2. The surface area to calculate includes: the lateral surface area of the rectangular prism (excluding the base and top), the lateral surface area of the cylinder (excluding the base and top), and the top face of the prism (which is covered by the cylinder's base but not exposed). However, since the cylinder is attached and its base is not exposed, only the outer surfaces are counted.
  3. The lateral surface area of a rectangular prism is given by 2h(l + w), where h is height, l and w are length and width. Here, l = 82 cm, w = 82 cm, h = 41 cm.
  4. Substitute values: 2 × 41 × (82 + 82) = 82 × 164. Calculate: 82 × 164 = 82 × (160 + 4) = 82×160 + 82×4 = 13120 + 328 = 13448 cm².
  5. The lateral surface area of a right circular cylinder is 2πrh, where r is radius and h is height. Here, r = 41 cm, h = 79 cm.
  6. Substitute values: 2 × π × 41 × 79. First compute 2 × 41 × 79 = 82 × 79. Calculate: 82 × 79 = 82 × (80 - 1) = 82×80 - 82 = 6560 - 82 = 6478. So lateral area = 6478π cm².
  7. The top face of the prism is covered by the cylinder's base, so it is not exposed. The base of the cylinder is not exposed because it is attached to the prism. Therefore, no additional face area is added.
  8. The base of the prism is not exposed (it rests on the floor), so it is excluded. The interface between prism and cylinder is not exposed, so it is excluded.
  9. Total surface area = lateral area of prism + lateral area of cylinder = 13448 + 6478π cm².
  10. The problem asks for the exact total surface area as a single expression in terms of π. The expression 13448 + 6478π is already in simplest form.
  11. Check: 82×164 = 13448 (correct), 2×41×79 = 6478 (correct), and no face is double-counted or omitted. The interface and base are excluded as required.
  12. The total surface area of one storage bin, excluding the base and interface, is 13448 + 6478π square centimetres.

Reference answer: 13448 + 6478π cm²

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