Lombard Street in San Francisco is a switchback road with a speed limit of 8km/hr because if it’s extreme grade. The angle of elevation of Lombard Street in one section reaches 15°. Determine the grade of this section of road.
Please help. I’ve tried drawing out diagram but I am stuck . Thank you!
the grade is the tangent of the angle.
tan15° = 0.2679 or about 27%
pretty steep!
Thank you for your help :)
To determine the grade of the section of Lombard Street with a 15° angle of elevation, we can use the tangent function.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the change in elevation and the adjacent side is the horizontal distance traveled.
Let's assume the change in elevation is represented by "x" and the horizontal distance traveled is represented by "y".
We know that the tangent of 15° is equal to the change in elevation divided by the horizontal distance traveled:
tan(15°) = x/y
To solve for x/y, we can rearrange the equation:
x/y = tan(15°)
Now we can substitute the values:
x/y = tan(15°)
But we need to find the grade, which is represented as a percentage. So we multiply the equation by 100:
(100x/y) = 100tan(15°)
The grade is equal to (100x/y), so we can rewrite the equation as:
Grade = 100tan(15°)
Now we can calculate the grade using a calculator:
Grade = 100 * tan(15°)
≈ 26.79%
Therefore, the grade of the section of Lombard Street with a 15° angle of elevation is approximately 26.79%.
To determine the grade of Lombard Street in the given section, we need to find the ratio of the vertical rise to the horizontal run.
First, let's define the terms:
- Vertical rise: The change in elevation or the height difference between the two points.
- Horizontal run: The horizontal distance between the two points.
From the information given, the angle of elevation of Lombard Street in the section is 15°. We can use trigonometry to find the grade.
Consider the triangle formed by the road, where the angle of elevation is 15° and the vertical rise is the height difference.
Let's label the sides of the triangle:
- Opposite side (height): Vertical rise
- Adjacent side (distance along the road): Horizontal run
In trigonometry, the tangent function relates the opposite side and the adjacent side of the right triangle:
Tangent(angle) = Opposite / Adjacent
In this case, we have:
Tangent(15°) = Vertical rise / Horizontal run
We can rearrange the equation to solve for the grade:
Grade = Vertical rise / Horizontal run = Tangent(15°)
To find the grade, we need to calculate the tangent of 15°. You can use a scientific calculator or an online trigonometric calculator to find the tangent of 15°. The tangent of 15° is approximately 0.267949.
Therefore, the grade of the section of Lombard Street with an angle of elevation of 15° is 0.267949 (or approximately 0.27).