NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Three-dimensional coordinates

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Required skills (click to reveal)

A craft designer in Melbourne is creating a 3D model of a decorative geometric sculpture using a digital design tool that employs a three-dimensional coordinate system. The sculpture's central support point is located at coordinates (23, 0, 81).

The designer needs to determine the exact distance from this central point to the origin (0, 0, 0) in the design space. Using the 3D distance formula: distance = sqrt(x^2 + y^2 + z^2), what is the exact distance in units?

Provide the answer as a simplified radical expression.

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Three perpendicular coordinate directions. A schematic projection of perpendicular x,y,z axes. The point (2,3,6) is reached by changes 2 in x, 3 in y and 6 in z. Perspective spacing is illustrative, not an angle or distance measurement.xyz(2,3,6)236
A schematic projection of perpendicular x,y,z axes. The point (2,3,6) is reached by changes 2 in x, 3 in y and 6 in z. Perspective spacing is illustrative, not an angle or distance measurement.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The central support point of the sculpture is located at coordinates (23, 0, 81) in a 3D coordinate system.
  2. The origin of the coordinate system is (0, 0, 0). The task is to find the straight-line distance from (23, 0, 81) to (0, 0, 0).
  3. The 3D distance formula is used: distance = sqrt(x^2 + y^2 + z^2), where x, y, and z are the differences in coordinates between the two points.
  4. Substitute the coordinates into the formula: x = 23, y = 0, z = 81.
  5. Calculate x^2: 23^2 = 529.
  6. Calculate y^2: 0^2 = 0.
  7. Calculate z^2: 81^2 = 6561.
  8. Sum the squares: 529 + 0 + 6561 = 7090.
  9. Take the square root of 7090: sqrt(7090).
  10. Factor 7090 to simplify the radical: 7090 = 10 × 709. Check if 709 is prime or has square factors. 709 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29. 29^2 = 841 > 709, so 709 is prime. Thus, no square factors other than 1.
  11. Therefore, sqrt(7090) cannot be simplified further. The exact distance is sqrt(7090) units.
  12. The distance from the central point (23, 0, 81) to the origin (0, 0, 0) is exactly sqrt(7090) units.

Reference answer: sqrt(7090)

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