Question

According to the Parking Standards in Santa Clarita, California, an access ramp to a parking lot cannot have a slope exceeding \(11^{\circ}\). Suppose a parking lot is 10 feet above the road. If the length of the ramp is 60 feet, does this access ramp meet the requirements of the code? Explain your reasoning.

Short answer

Yes, the ramp's slope is approximately \(9.59^{\circ}\), which is less than \(11^{\circ}\).

Step-by-step solution

Understand the Problem

We need to find out if the slope of the ramp to the parking lot exceeds the allowed slope of \(11^{\circ}\). The ramp rises 10 feet over a horizontal distance of 60 feet.

Recall the Trigonometric Concept

The slope of a ramp can be determined using the sine of the angle of inclination. The formula is \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), where the opposite side is the rise (10 feet) and the hypotenuse is the length of the ramp (60 feet).

Calculate the Sine of the Angle

Using the formula \( \sin(\theta) = \frac{10}{60} \), we calculate \( \sin(\theta) = \frac{1}{6} \approx 0.1667 \).

Determine if the Slope Meets the Requirement

We need to check if \( \sin(11^{\circ}) \) is greater than \( \frac{1}{6} \). The sine of \(11^{\circ}\) is approximately 0.1908. Since 0.1908 is greater than 0.1667, the slope of the ramp at \( \approx 9.59^{\circ} \) is less than \(11^{\circ}\).

Conclusion

Since the calculated angle of the ramp's slope is less than \(11^{\circ}\), the ramp meets the requirements of the Santa Clarita parking code.

Key concepts

Slope Calculation

When discussing slopes, think of them as the steepness or incline of a line or surface from the horizontal. For ramps, the key is calculating how high they rise over a given horizontal distance. In the exercise provided, the rise is 10 feet, and the horizontal length of the ramp is 60 feet. Together, these form the triangle we use to calculate the slope.

To determine the slope, a common approach is using the rise over the run. In this particular exercise:

• The rise is 10 feet.
• The length of the ramp (acting as the hypotenuse) is 60 feet.

By dividing the rise by the run, we essentially find the tangent of the angle of inclination, often expressed as the ratio \( \frac{10}{60} = \frac{1}{6} \). This fraction gives us a measure of how steep the ramp is, indicating that for every 6 feet horizontally, the ramp rises by 1 foot.

Angle of Inclination

The angle of inclination refers to the angle that forms between a ramp or slope and the horizontal ground. In practical terms, it tells us how slanted or steep a surface is, which is crucial for accessibility and compliance with safety standards.

To find the angle of inclination, we use trigonometry. The sine function is particularly useful since it relates the angle to the ratio of the rise over the hypotenuse:

• The sine function (\( \sin(\theta) \)) is the opposite side (the rise in feet) divided by the hypotenuse (the length of the ramp).
• In this case, \( \sin(\theta) = \frac{10}{60} = 0.1667 \).

By comparing this with the sine of the maximum allowable angle, we decide if the slope meets set standards. For this exercise, the ramp's angle is approximately \( 9.59^{\circ} \), which is less than the maximum allowed \( 11^{\circ} \). This suggests that the ramp complies with safety codes, as its angle of inclination falls within acceptable limits.

Trigonometric Functions

Trigonometry often comes into play with slopes and angles, especially when considering real-world applications like ramps. The trigonometric functions—sine, cosine, and tangent—help us understand and compute these angles and dimensions effectively from given parameters.

In this exercise, the sine function is particularly in focus, since:

• It correlates the angle of inclination to the relative lengths of a ramp.
• Here, \( \sin(\theta) = \frac{\text{rise}}{\text{hypotenuse}} \), or \( \frac{10}{60} \).

This ratio, when converted to degrees, offers a clear picture of the slope's steepness. Checking this against known trigonometric values of standard angles, like \( \sin(11^{\circ}) = 0.1908 \), solidifies our understanding of whether this slope is suitable according to building codes.

Understanding these functions allows for accurate predictions and verifications against regulations, making trigonometry an essential tool in architectural and construction calculations.