Slope in Geometry Math Example 2

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Example 2

hard
A road rises 4040 metres over a horizontal distance of 500500 metres. Express the slope as a percentage grade and as an angle (to the nearest tenth of a degree).

Solution

  1. 1
    Step 1: Slope =riserun=40500=0.08= \dfrac{\text{rise}}{\text{run}} = \dfrac{40}{500} = 0.08.
  2. 2
    Step 2: Percentage grade =0.08×100%=8%= 0.08 \times 100\% = 8\%.
  3. 3
    Step 3: Angle θ=arctan⁡(0.08)≈4.6°\theta = \arctan(0.08) \approx 4.6° (using a calculator: tan⁡−1(0.08)≈4.574°\tan^{-1}(0.08) \approx 4.574°, round to 4.6°4.6°).

Answer

Slope =0.08= 0.08; grade =8%= 8\%; angle ≈4.6°\approx 4.6°.
Slope is the ratio rise/run. In engineering, roads are described by percentage grade (slope ×100%\times 100\%). The angle of inclination is found via arctan⁡(m)\arctan(m). An 8%8\% grade is considered steep for a highway.

About Slope in Geometry

The steepness of a line expressed as rise over run, connecting the algebraic slope formula to the geometric angle of inclination.

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More Slope in Geometry Examples

Example 1 easy

Find the slope of the line through [formula] and [formula]. Describe what the slope tells us about t

Example 3 easy

Which of these lines is steeper: [formula] or [formula]? Which has a greater angle with the [formula

Example 4 medium

Points [formula], [formula], [formula] are collinear. Find [formula].

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