NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Volume of prisms

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A library in Melbourne stores books in rectangular prism-shaped crates. Each crate has a base that measures 89 cm by 40 cm and a height of 27 cm.

The volume of a rectangular prism is calculated using the formula V = (area of base) × (height).

Calculate the exact volume of one crate in cubic centimetres, expressing your answer as a simplified fraction.

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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The crate is a rectangular prism with base dimensions 89 cm and 40 cm, and height 27 cm. The volume of a rectangular prism is given by V = (area of base) × (height).
  2. Calculate the area of the base: area = length × width = 89 cm × 40 cm.
  3. 89 × 40 = (80 + 9) × 40 = 80×40 + 9×40 = 3200 + 360 = 3560 cm².
  4. Now multiply the base area by the height: V = 3560 cm² × 27 cm.
  5. Compute 3560 × 27 using distributive property: 3560 × (20 + 7) = 3560×20 + 3560×7.
  6. 3560 × 20 = 3560 × 2 × 10 = 7120 × 10 = 71200.
  7. 3560 × 7 = (3500 + 60) × 7 = 3500×7 + 60×7 = 24500 + 420 = 24920.
  8. Add the two products: 71200 + 24920 = 96120.
  9. The volume is 96120 cm³. This is an integer, so as a fraction it is 96120/1.
  10. Check if 96120/1 can be simplified. Since the denominator is 1, the fraction is already in simplest form.
  11. The volume of one crate is exactly 96120 cm³, expressed as a simplified fraction.
  12. Final verification: 89 × 40 × 27 = 89 × 1080 = 89 × (1000 + 80) = 89000 + 7120 = 96120. Correct.

Reference answer: 96120

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