The steepness, or grade, of a highway or railroad is expressed as a percent. In the photo of the cog railway at Pike's Peak, in Colorado, the grade is 18%. Thus, for every 100 ft of horizontal run, the train rises 18 ft. Find the angle of the inclination of the railway.

Question

Tamara Warren

Mathematics

Today, 02:19

The steepness, or grade, of a highway or railroad is expressed as a percent. In the photo of the cog railway at Pike's Peak, in Colorado, the grade is 18%. Thus, for every 100 ft of horizontal run, the train rises 18 ft. Find the angle of the inclination of the railway.

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Carlos Kirby · Answer 1

The angle of inclination is 10.2°.

The angle of inclination will be between the hypotenuse of the triangle and the bottom leg. The bottom leg is the 100 ft run. The rise of 18 feet will be the other leg, which is opposite this angle.

This gives us the sides opposite and adjacent the angle we are looking for; opposite/adjacent is the ratio for tangent:

tan x = 18/100

We can solve this using inverse tangent:

tan⁻¹ x = tan ⁻¹ (18/100)

x = 10.2°