HOW TO MEASURE THE SLOPE OF A ROAD
Land
surveyors use trigonometry and their fancy equipment to measure things like the
slope of a piece of land (how far it drops over a certain distance).
Have you
ever noticed a worker along the road, peering through an instrument, looking at
a fellow worker holding up a sign or a 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.
You may
recognize that 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 a length or distance.
To solve one
of these Surveying 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 —
voile!
Suppose that
Elliott and Fred are making measurements for the road-paving crew. They need to
know how much the land slopes downward along a particular stretch of road to be
sure there’s proper drainage.
Elliott
walks 80 feet from Fred and holds up a long pole, perpendicular to the ground,
that has markings every inch along it. Fred looks at the pole through a
sighting instrument. Looking straight across, parallel to the horizon, Fred
sights a point on the pole 50 inches above the ground — call it point A. Then
Fred looks through the instrument at the bottom of the pole, creating an angle
of depression.
What is the angle of depression, or slope of the Road, to where Elliott 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 insert the values that you know.
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. In this example, convert feet to inches.
80 feet = 80 x 12 inches = 960 inches
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Or
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Substituting in the values, you get the tangent of some angle with a measure of x degrees:
Solve for the
value of x.
In the
Appendix, you see that 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 Elliott and Fred.
THANKS
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