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Chapter 6: Problem 84

A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

Short Answer

Expert verified
The slope of the road is equal to tangent of 0.20 radians, and the change in elevation over one mile is this slope times 1609.34 meters.

Step by step solution

01

Calculating the Slope of the Road

In mathematics, the slope of a line is equal to the tangent of the angle of incline, which in this context is the angle the road makes with the horizontal. We find this by taking the tangent of the given angle:\n\n\( \text{{Slope}} = \tan(0.20) \)
02

Calculate the Change in Elevation

The 'change in elevation' refers to how far 'up' you go while traveling along the road. In terms of our triangle, this is equivalent to finding the 'opposite' side, using the formula for the tangent of an angle (which is opposite over adjacent). Given the distance of one mile (or 1609.34 meters) and using the slope derived in the previous step, we can calculate the vertical distance or change in elevation: \n\n \( \text{{Change in elevation}} = \text{{Slope}} \times \text{{distance}} \)

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