Subject: change in elevation
Name: Nancy
Who are you: Parent

If a portion of highway has a grade or slope of 12%, and the length of this portion is approximately 1.1 miles, what is the change in elevation from the top of the grade to the bottom of the grade in feet?


Hi Nancy.

The "Grade" of an incline or decline in a highway is just the ratio of the rise (or drop) over the run (horizontal distance) expressed as a percentage. For example, if a road drops by 100 meters over a horizontal distance of 1 km, then its grade is -10%.

The important picture to create is one of a right-angled triangle with the hypotenuse representing the roadway, the horizontal is the "run" and the vertical side is the "rise".

With your question, you know the grade and the length of the hypotenuse (the roadway) and are asked for the change in elevation.

Use basic trigonometry to calculate this. It is the length of the short side of the right triangle. The grade written as a fraction, rather than a percentage, is the same as the tangent of the angle (because tangent is rise over run as well). So you can use arctangent to calculate the angle. Now with the angle and the hypotenuse, you can use sine to calculate the rise.

Hope this helps,
Stephen La Rocque.

Hi Nancy,

I read your problem slightly differently than Stephen La Rocque. A 12% grade for a road is quite steep but still the triangle Sue asks you to picture very long and thin and thus the long side is approximately the same length as the hypotenuse. Since you know that the portion of the road being considered is approximately 1.1 miles I see the length of the long leg of the triangle as approximately 1.1 miles. Since the grade is 12% the height of the rise in the road is approximately 12% of 1.1 miles.