- Model the ramp as a right-angled triangle where the ramp length is the hypotenuse (\(8 \text{ ft}\)) and the rise is the opposite side (\(5 \text{ ft}\)).
- Recognize that the sine of the angle of elevation \(\theta\) is the ratio of the opposite side to the hypotenuse, \(\sin(\theta) = \frac{5}{8}\).
- Solve for \(\theta\) by taking the inverse sine, resulting in \(\theta = \arcsin(\frac{5}{8})\).
- The final answer is \(\boxed{\arcsin \left(\frac{5}{8}\right)}\).
Explanation
- Data and problem analysis
The problem describes a ramp that can be modeled as a right-angled triangle. We are given:
- The length of the ramp, which is the hypotenuse of the triangle: \(c = 8 \text{ ft}\).
- The height of the rise, which is the side opposite to the angle of elevation: \(a = 5 \text{ ft}\).
- The objective is to find the angle of elevation, \(\theta\), which is the angle between the ramp and the horizontal ground.
- Relating sides to trigonometric ratios
In a right-angled triangle, the primary trigonometric ratios are defined based on the sides relative to an angle \(\theta\). Specifically:
- \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
- \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
- \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
Given that we have the opposite side (5 ft) and the hypotenuse (8 ft), the sine function is the most direct ratio to relate these values to the angle of elevation \(\theta\):
\[\sin(\theta) = \frac{5}{8}\]
- Solving for the angle of elevation
To solve for the angle of elevation \(\theta\), we use the inverse sine function (also known as the arcsine function):
\[\theta = \arcsin\left(\frac{5}{8}\right)\]
This expression represents the angle whose sine is \(\frac{5}{8}\). Comparing this result with the given options, we identify it as the first choice provided.
Examples
In construction and engineering, determining the angle of elevation for a ramp is crucial for safety and accessibility, such as for wheelchair ramps. A standard for accessibility often requires the slope not to exceed a certain ratio (like 1:12). Knowing the length and the rise allows you to calculate the precise angle \(\theta = \arcsin(\frac{\text{rise}}{\text{length}})\) to ensure it complies with building codes and provides a comfortable incline.