Problem 65 - Algebra and Trigonometry

Question

Grade of a Road West of Albuquerque, New Mexico, Route 40 eastbound is straight and makes a steep descent toward the city. The highway has a 6\(\%\) grade, which means that its slope is \(-\frac{6}{100}\) . Driving on this road, you notice from elevation signs that you have descended a distance of 1000 ft. What is the change in your horizontal distance?

Step-by-Step Solution

Answer

The change in horizontal distance is approximately 16666.67 feet.

Step 1: Understand the Definition of Grade

The grade of a road gives the slope, expressed as a percentage. In this problem, a 6% grade means that for every 100 feet horizontally, the elevation decreases by 6 feet. Thus, the slope is a ratio of rise over run and is given as \[-\frac{6}{100}.\]

Step 2: Identify Known Values

We are given the road’s grade (slope) as \(-\frac{6}{100}\) and the vertical change (descent in elevation) as 1000 feet. We need to find the horizontal distance corresponding to this descent.

Step 3: Set Up the Equation

The slope formula is \[\text{slope} = \frac{\text{vertical change}}{\text{horizontal change}}.\] Rearrange to find horizontal change:\[\text{horizontal change} = \frac{\text{vertical change}}{\text{slope}}.\] Substitute the known values:\[\text{horizontal change} = \frac{1000}{\frac{6}{100}}.\]

Step 4: Solve the Equation

Compute the horizontal change by performing the division:\[\text{horizontal change} = \frac{1000}{\frac{6}{100}} = 1000 \times \frac{100}{6} = \frac{100000}{6} \approx 16666.67.\] This represents the horizontal distance covered.

Key Concepts

Grade of a Road

The term "grade of a road" refers to the steepness or incline of a road's surface. It's often expressed as a percentage, indicating how much the road rises or falls over a certain horizontal distance. For example, a 6% road grade means that for every 100 feet you travel horizontally, the road changes its elevation by 6 feet either up or down.
This concept is essential for understanding how slopes affect driving conditions, construction, and road safety. A steeper road grade might require slower speeds for vehicles or additional engineering to manage water runoff and erosion.
The slope, or grade, can be positive if the road is going uphill, or negative if it's going downhill. In our example with Route 40, the descent is indicated by a negative grade of -\(\frac{6}{100}\)\.-Understanding the grade helps drivers anticipate how their vehicles will perform when climbing or descending.

Percentage Grade

The percentage grade is a way of describing a road's slope using a simple percentage. Rather than dealing with fractions or slopes in decimal form, percentage grade is easy to understand because it directly relates to the change in elevation over a certain distance.
In the context of roads, a grade of 6% means that there is a 6-foot vertical drop for every 100 feet traveled horizontally.

• This is written as \(-\frac{6}{100}\), which is useful in calculations as a ratio.
• This percentage allows easier communication and understanding of the steepness without confusing fractions or decimals.

Using percentage grades helps simplify planning and communication for engineers and city planners when designing roads. It's crucial for both structural considerations and ensuring that the road is safe and drivable under various conditions.

Horizontal Distance Calculation

Calculating horizontal distance from a given slope is an application of basic geometric principles. You start by understanding the relationship between vertical change (rise or fall) and horizontal change (run). The formula \[\text{slope} = \frac{\text{vertical change}}{\text{horizontal change}} \]is the foundational equation for these calculations.

• Given the slope and the vertical change (e.g., elevation drop), you can rearrange this equation to solve for horizontal change.\[\text{horizontal change} = \frac{\text{vertical change}}{\text{slope}}\]
• In our Route 40 example, with a vertical change of 1000 feet and a slope of \(-\frac{6}{100}\), you calculate horizontal distance as:\[\text{horizontal change} = \frac{1000}{\frac{6}{100}} = 16666.67 \text{ feet}\]

This calculation provides the actual distance traveled horizontally, which can be much longer than the vertical drop, especially in hilly or mountainous terrain.

Vertical Change

Vertical change refers to how much the elevation changes when traveling along a particular path or road. This metric is vital as it influences everything from driving behavior to water drainage and road design.
In the case of road grades, the vertical change is often described in terms of feet or meters over a traveled horizontal distance. For a road with a grade, you should always measure this change to understand how a vehicle will descend or climb.

• For Route 40 in our example, the vertical change is 1000 feet.
• Understanding this measurement, combined with the road's slope, allows you to calculate the horizontal distance using ratios.

In practical terms, vertical change affects how vehicles operate, including necessary power for climbs and braking effectiveness on descents. Road designers must consider these factors to ensure safety and efficiency on the roads.