Problem 61 - Algebra and Trigonometry

Question

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 horizontal change is 16666.67 feet.

Step 1: Understanding the Grade

The grade of the road is given as 6\(\%\), meaning that for every 100 horizontal feet along the road, you descend 6 feet vertically. This information gives you the relationship between vertical and horizontal distances on the road.

Step 2: Setting Up the Relationship

Since the slope or grade is given as \(-\frac{6}{100}\), this can be expressed as \(-0.06\). The negative sign indicates a descent or downward movement in the vertical direction as you move forward horizontally.

Step 3: Using the Formula for Slope

The formula for slope \(m\) is given by the change in vertical distance (\(\Delta y\)) over the change in horizontal distance (\(\Delta x\)). So \(m = \frac{\Delta y}{\Delta x} = -0.06\).

Step 4: Solving for Horizontal Change

Rearrange the equation \(\frac{\Delta y}{\Delta x} = -0.06\) to solve for \(\Delta x\): \(\Delta x = \frac{\Delta y}{-0.06}\). Here, \(\Delta y = -1000\) ft (negative due to descent). Substitute this value in: \(\Delta x = \frac{-1000}{-0.06}\).

Step 5: Calculating the Change

Perform the division: \(\Delta x = \frac{-1000}{-0.06} = 16666.67\) ft. Therefore, the change in horizontal distance is 16666.67 feet.

Key Concepts

Slope Calculation

When tackling road problems, one key concept is slope calculation. The slope represents how steep a line or surface is, in our case, a road. It's essentially a measure of the angle of incline or decline. The mathematical formula for calculating slope is:

• \( m = \frac{\Delta y}{\Delta x} \)

Here, \(\Delta y\) is the change in vertical distance, while \(\Delta x\) is the change in horizontal distance. For roads, particularly, this formula helps determine the relationship between elevation changes and length traveled.
A negative slope indicates a descent, meaning the road is going downwards as you travel forward. To make slope calculation simple:

• Identify the vertical and horizontal changes.
• Plug these values into the slope formula.
• Solve for the desired variable if needed.

Understanding this concept helps in analyzing road inclines and declines, which is useful for driving or highway construction.

Grade of a Road

The grade of a road, often expressed as a percentage, is a specific way of describing a road's slope. It's actually comparing the rise (vertical) over the run (horizontal) to describe how steep a road is. For example, a 6\(\%\) grade means that for every 100 feet of horizontal distance, the road changes elevation by 6 feet.
In the context of highways and roads, the grade helps drivers know what to expect in terms of incline or decline.

Why is Grade Important?

The grade is crucial for several reasons:

• Safety: Knowing the grade can help drivers adjust speed and braking strategies, especially in steep descents or ascents.
• Construction: Engineers use grade percentages to design roads for maximum safety and efficiency.
• Merchandise Transport: Trucks carrying heavy loads must consider road grades to manage speed and delivery times.

Grasping this concept is vital in understanding how slope impacts driving conditions.

Change in Distance

Analyzing the change in distance involves understanding the relationship between elevation change and horizontal distance traveled. Using the slope formula, you can solve for the missing variable, which, in many cases, could be the horizontal change \(\Delta x\).
In practice, the calculation might look like this: if a road descends 1000 feet vertically, and we know the slope (as a negative percentage), we can find out the horizontal distance the descent covers using:

• \( \Delta x = \frac{\Delta y}{slope} \)

Here, \(\Delta y\) is negative, representing a descent, and once you input values, you will calculate the length of the road needed to achieve this drop in height.

• This method helps in planning road trips and constructions by giving clear distance evaluations.
• It is also useful for adjusting driving strategies based on road conditions and expected travel lengths.

This understanding helps apply mathematical principles practically to real-world scenarios such as road travel.