NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Surface area and volume

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A student in Brisbane is designing a cylindrical can-cooler made from a single piece of fabric that wraps around a right circular cylinder with a height of 97 cm and a base radius of 84 cm. The cooler must cover the lateral surface only, and the fabric is cut from a rectangular sheet with no waste.

What is the exact surface area of the lateral surface of the cylinder, in square centimetres, expressed in terms of π? Use the formula for lateral surface area of a cylinder: SA = 2πrh, where r is the radius and h is the height.

Provide the exact answer as a simplified expression involving π.

Circle measurementsradius = 84 cmr = 84 cm
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Concept illustrations

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A rectangular prism. A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.4 cm3 cm2 cm
A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.
Square the length conversion factor. A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.1 m = 100 cm1 m100 cm
A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.

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Formula and explanation (click to reveal)

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The problem requires finding the lateral surface area of a right circular cylinder with height h = 97 cm and base radius r = 84 cm.
  2. The lateral surface area of a cylinder is given by the formula SA = 2πrh, which applies because the surface is curved and wraps around the cylinder without top or bottom.
  3. Substitute the given values into the formula: SA = 2 × π × 84 × 97.
  4. First, compute the product of the constants: 2 × 84 × 97.
  5. Calculate 2 × 84 = 168.
  6. Now compute 168 × 97. Use distributive property: 168 × (100 - 3) = 168×100 - 168×3.
  7. 168 × 100 = 16,800.
  8. 168 × 3 = 504.
  9. Subtract: 16,800 - 504 = 16,296.
  10. Now include π: SA = 16,296π cm².
  11. The expression 16,296π is already simplified and exact, as no common factors exist between 16,296 and π.
  12. The lateral surface area of the cylinder is exactly 16,296π square centimetres.

Reference answer: 16296π cm²

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