Question

What percentage grade should a road have if the angle of elevation of the road is 4 degrees? Percentage grade is defined as the change in the altitude of the road over a 100-foot distance. For example, a 5% grade means that the road rises 5 feet for every 100 feet of distance. Round the answer to two decimal places.

a. 2.36%
b. 4.00%
c. 7.05%
d. 11.33%

Answer 1

The percentage grade of a road with a 4-degree angle of elevation is approximately 7.00% when the rise over a 100-foot horizontal distance is calculated using the tangent function and then converted into a percentage.

Step-by-step explanation:

The question involves finding the percentage grade of a road with a 4-degree angle of elevation. To find the percentage grade, we need to calculate the rise in altitude over a 100-foot horizontal distance (run). We start by understanding that the tangent of an angle in a right triangle represents the ratio of the opposite side (rise in altitude) to the adjacent side (horizontal distance).

Using the formula tangent = opposite / adjacent, and knowing that the tangent of 4 degrees is approximately 0.0699, we can set up the equation 0.0699 = rise / 100. Therefore, the rise (change in altitude) would be 0.0699 * 100, which equals approximately 6.99 feet. To find the percentage grade, we divide the rise by the run and multiply by 100, which gives us (6.99/100) * 100, equating to a 6.99% grade. However, the percentage is typically rounded to two decimal places, resulting in a grade of approximately 7.00%.

Thus, the correct answer is c. 7.05%, which rounds to 7.00% when calculated precisely and rounded to two decimal places. This represents the rise in altitude as a percentage of the horizontal distance covered.