NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Trigonometry, bearings and elevation

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Required skills (click to reveal)

A school in Darwin is ordering a new set of science lab equipment. The delivery truck travels 86 m east and then 26 m north from the main gate to reach the science block's storage area.

A student on the roof of the science block, located directly above the storage area, observes the truck at an angle of elevation of 45 degrees from the ground. Using trigonometry and the given distances, determine the exact height of the student's observation point above the ground in metres.

The answer must be given as an exact expression involving a radical or fraction, not a decimal.

Concept illustrations

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Choose the angle before naming sides. For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.opp. 3adj. 4hyp. 5θsin θ = 3/5
For angle θ at the lower-right vertex, opposite = 3, adjacent = 4 and hypotenuse = 5. Therefore sin θ = 3/5. The triangle is right-angled.
Bearings are clockwise from north. The route makes a 060° clockwise turn from north. The opposite route has bearing 240°. Compass directions are shown; the diagram is schematic.NESW060°240°
The route makes a 060° clockwise turn from north. The opposite route has bearing 240°. Compass directions are shown; the diagram is schematic.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The delivery truck travels 86 m east and 26 m north from the main gate to reach the storage area. These two distances form the legs of a right-angled triangle, with the straight-line distance from the main gate to the storage area as the hypotenuse.
  2. The student is on the roof directly above the storage area, so the horizontal distance from the main gate to the student's position is the same as from the main gate to the storage area. This horizontal distance is the hypotenuse of the triangle formed by the 86 m and 26 m legs.
  3. Use Pythagoras’ theorem to find the horizontal distance from the main gate to the student's position: d = sqrt(86^2 + 26^2).
  4. Calculate 86^2 = 7396 and 26^2 = 676.
  5. Add the squares: 7396 + 676 = 8072.
  6. Take the square root: d = sqrt(8072).
  7. The student observes the truck at an angle of elevation of 45 degrees. In a right-angled triangle, if the angle of elevation is 45 degrees, then the opposite side (height) equals the adjacent side (horizontal distance from observer to point on ground).
  8. Since tan(45°) = 1, and tan(θ) = opposite / adjacent, it follows that height = horizontal distance from student to truck's ground position.
  9. The horizontal distance from the student to the truck is the same as the straight-line distance from the main gate to the storage area, which is sqrt(8072) m.
  10. Therefore, the height of the student's observation point is exactly sqrt(8072) metres.
  11. Simplify sqrt(8072). Factor 8072: 8072 = 4 × 2018 = 4 × 2 × 1009. 1009 is a prime number. So sqrt(8072) = sqrt(4 × 2 × 1009) = 2 × sqrt(2018).
  12. The height is exactly 2*sqrt(2018) metres. This is the simplest exact radical form.

Reference answer: 2*sqrt(2018) m

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