NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Two-dimensional representations

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Required skills (click to reveal)

A three-dimensional model of a structure in Melbourne is composed of a rectangular prism base measuring 9 m in length, 5 m in width, and 4 m in height, with a right triangular prism roof attached to the longer side. The triangular roof has a base equal to the length of the base (9 m) and a height of 3 m.

When viewed from the front (looking at the 9 m by 4 m face), the two-dimensional representation shows a composite shape consisting of a rectangle and a triangle on top.

What is the exact area, in square metres, of this two-dimensional front view?

Concept illustrations

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The front view of the structure consists of a rectangle and a right triangle stacked vertically. The rectangle represents the front face of the base, which is 9 m long and 4 m high.
  2. The area of the rectangular part is calculated using the formula: area = length × height. Substituting the values: area = 9 m × 4 m.
  3. Performing the multiplication: 9 × 4 = 36. The area of the rectangle is 36 square metres.
  4. The triangular part of the front view is a right triangle with base 9 m (same as the length of the base) and height 3 m (the height of the triangular roof).
  5. The area of a triangle is given by: area = (1/2) × base × height. Substituting the values: area = (1/2) × 9 m × 3 m.
  6. First, multiply 9 × 3 = 27. Then, multiply by 1/2: (1/2) × 27 = 13.5. The area of the triangle is 13.5 square metres.
  7. The total area of the front view is the sum of the rectangular and triangular areas: 36 m² + 13.5 m².
  8. Adding the areas: 36 + 13.5 = 49.5. The total area is 49.5 square metres.
  9. The front view is a composite shape made of a rectangle and a triangle. The representation is valid because the structure's front face includes both the vertical wall and the sloped roof.
  10. The result is exact: 49.5 m² is equivalent to 99/2 m², but since the question asks for the exact area and 49.5 is a terminating decimal, it is acceptable as a decimal.
  11. The front view area is 49.5 square metres, which matches the sum of the rectangle and triangle areas.
  12. The final answer is 49.5 square metres, as the sum of the two-dimensional components of the front face.

Reference answer: 49.5

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