Question

The angle of inclination of a road is often expressed as a percent grade, which is the vertical rise divided by the horizontal run( expressed as a percent). A 9.0% grade corresponds to a road that rises 9.0 ft for every 100 ft along the horizontal. Find the angle of the inclination that corresponds to a 9.0% grade.(Round to the nearest tenth)

Answer 1

The angle of inclination for a 9.0% grade is found by taking the inverse tangent of 0.09, which gives us an angle of approximately 5.1 degrees when rounded to the nearest tenth.

Step-by-step explanation:

To find the angle of inclination that corresponds to a 9.0% grade, we need to understand that the percent grade represents the tangent of the angle of inclination. The 9.0% grade means there is a rise of 9.0 feet for every 100 feet of horizontal distance. In mathematical terms, this can be expressed as tan(θ) = Δy / Δx, where θ is the angle of inclination, Δy is the rise, and Δx is the run. For a 9.0% grade, the tangent of the angle is 9/100, which is 0.09.

To find the angle, we must take the inverse tangent (arctan) of 0.09:

θ = arctan(0.09)

Using a calculator to find the inverse tangent, we get:

θ ≈ 5.1°

Therefore, the angle of inclination that corresponds to a 9.0% grade, when rounded to the nearest tenth, is approximately 5.1 degrees.

Answer 2

Explanation:

A 9.0% grade corresponds to a road that rises 9.0 ft for every 100 ft along the horizontal.

• Let the angle of inclination that corresponds to a 9.0% grade = x

The diagram below is a representation of the problem:

Using trigonometrical ratios, we find the value of angle x.

`\begin{gathered} \tan\theta=(Opposite)/(Adjacent) \\ \implies\tan x=(9)/(100) \end{gathered}`

Take the arctan of both sides:

`\begin{gathered} x=arctan(0.09) \\ x=5.1\degree \end{gathered}`

The angle of the inclination that corresponds to a 9.0% grade is 5.1 degrees.