Problem 59 - Algebra for College Students

Question

A certain highway has a \(2 \%\) grade. How many feet does it rise in a horizontal distance of 1 mile? (1 mile \(=\) 5280 feet)

Step-by-Step Solution

Answer

The vertical rise is 105.6 feet.

Step 1: Understand the Meaning of a 2% Grade

A 2% grade means that for every 100 feet of horizontal distance, the road rises 2 feet vertically.

Step 2: Convert 1 Mile to Feet

Since 1 mile is equal to 5280 feet, we begin by confirming the horizontal distance as 5280 feet.

Step 3: Calculate the Vertical Rise

To find the vertical rise, multiply the horizontal distance (5280 feet) by the grade (0.02). Thus, the rise is \( 5280 \times 0.02 \).

Step 4: Perform the Multiplication

Calculate \( 5280 \times 0.02 = 105.6 \) feet.

Key Concepts

Vertical Rise

Understanding vertical rise is key to figuring out how much higher the road will be if you're moving along a certain slope or grade. In this context, vertical rise is the number of feet a road ascends over a horizontal distance. Vertical rise plays an important role in road and building design, as it affects the effort needed to go up or downhill. - Engineers often calculate vertical rise to ensure that roads are safe and accessible. - It's also useful for estimating how steep a path will be. Given a percentage grade and a specific horizontal distance, you can quickly determine the vertical rise. For instance, if a road rises 2 feet for every 100 feet traveled horizontally, this translates to a vertical rise as per the percentage grade.

Percentage Grade

The concept of percentage grade is a measure of the steepness of a road or path. Basically, it relates the amount of vertical rise to the horizontal distance traveled: - A 2% grade indicates that for every 100 feet traveled horizontally, there is a rise of 2 feet. - It's a simple way to express incline. When you see a percentage grade, think about it as a ratio: - A higher percentage grade means a steeper incline. - Roads or paths with a high grade can be more challenging for vehicles or pedestrians. Percentage grades are essential in planning and designing infrastructure, especially in hilly or mountainous areas, to ensure safety and efficiency.

Mile to Feet Conversion

Converting measurements is often necessary when solving real-world problems. A common conversion is from miles to feet, crucial in the United States where distances are often measured in miles, but construction and design might require feet. Here's the conversion you need to remember: - 1 mile equals 5280 feet. For practical applications: - Use this conversion to determine the total length in feet when only the distance in miles is known. - This is essential when calculating areas, volumes, or when needing more precise lengths. This conversion is particularly useful for map reading, travel planning, and when engaging in activities that require precise distance measurements, such as running or hiking.