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Chapter 6: Problem 65

A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

Short Answer

Expert verified
The slope of the road is approximately 0.10 or 10%. The change in elevation of the road over a two-mile stretch is approximately 1056 feet.

Step by step solution

01

Calculation of the slope of the road

The slope of the road is given by the tangent of the angle of inclination. For an angle measured in radians, the tangent of the angle can be approximated as the angle itself when the angle is small enough. Since 0.10 radian is a small angle, it is justifiable to assume that \( \tan(0.10) \approx 0.10 \). Hence the slope of the road is approximately 0.10 or 10%.
02

Calculation of the change in elevation over two miles

The change in elevation or the rise in the grade of the road is given by the product of the slope and the length of the road. So, for a two-mile stretch of road, the change in elevation is \(2 \, \text{miles} \times 0.10 = 0.2 \, \text{miles}\). Since 1 mile is approximately 5280 feet, the rise in the grade of the road is approximately \(5280 \times 0.2 = 1056 \, \text{feet}\).
03

Final results

Summarizing the calculations, the slope of the road is approximately 0.10 or 10%, and the rise in the grade of the road over a two-mile stretch is approximately 1056 feet.

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