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How to Measure the Slope of a Road

Trigonometry functions have everyday applications. For example, you can use a trigonometry function to measure the slope of the surface you are standing on.

Have you ever noticed a worker along the road, peering through an instrument, looking at a fellow worker holding up a sign or flag? Haven’t you ever wondered what they’re doing? Have you wanted to get out and look through the instrument, too? With trigonometry, you can do just what those workers do — measure distances and angles. Land surveyors use trigonometry and their fancy equipment to measure things like the slope of a piece of land.

The slope of land downward is sort of like an angle of depression. Slopes, angles of depression, and angles of elevation are all interrelated because they use the same trig functions. It’s just that in slope applications, you’re solving for the angle rather than being given it.

To solve problems involving slope, you can use the trig ratios and right triangles. One side of the triangle is the distance from one worker to the other; the other side is the vertical distance from the ground to a point on a pole. You form a ratio with those measures and determine the angle — voilà!

image0.jpg

Suppose that Dan and Stan are making measurements for the road-paving crew. They need to know how much the land slopes downward along a particular stretch of road. Dan walks 80 feet from Stan and holds up a long pole, perpendicular to the ground, that has markings every inch along it. Stan looks at the pole through a sighting instrument. Looking straight across, parallel to the horizon, Stan sights a point on the pole 50 inches above the ground — call it point A. Then Stan looks through the instrument at the bottom of the pole, creating an angle of depression. The preceding figure shows a diagram of this situation. What is the angle of depression, or slope of the road, to where Dan is standing?

  1. Identify the parts of the right triangle that you can use to solve the problem.

    The values you know are for the sides adjacent to and opposite the angle of depression. Call the angle measure x.

  2. Determine which trig function to use.

    The tangent of the angle with measure x uses opposite divided by adjacent.

  3. Write an equation with the trig function, then input the values that you know.

    image1.png

    In this problem, you need to write the equation with a common unit of measurement — either feet or inches. Changing 80 feet to inches makes for a big number; changing 50 inches to feet involves a fraction or decimal. Whichever unit you choose is up to you. This example converts feet to inches.

    80 feet = 80 12 inches = 960 inches

    Substituting in the values, you get the tangent of some angle with a measure of x degrees:

    image2.png
  4. Solve for the value of x.

    An angle of 2.9 degrees has a tangent of 0.0507, and a 3-degree angle has a tangent of 0.0524. The 3-degree angle has a tangent that’s closer to 0.05208333, so you can estimate that the road slopes at a 3-degree angle between Dan and Stan.

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