The percent grade of a highway is the amount that the highway raises or falls in a given horizontal distance. For example, a highway with a four percent grade rises 0.04 mile for every 1 mile of horizontal distance.
a. How many feet does a highway with a six percent grade rise in 2.5 miles of horizontal distance?
b. How many feet would a highway with a six percent grade rise in 90 miles of horizontal distance if the grade remained constant?
c. How is percent grade related to slope?
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To answer these questions, we need to understand the relationship between percent grade, slope, and how to calculate the rise in feet.
a. To calculate the rise in feet for a given distance with a percent grade, we can use the formula:
Rise (in feet) = Percent Grade (%) * Horizontal Distance (in miles) * 5280 (feet per mile)
For a highway with a six percent grade and a horizontal distance of 2.5 miles:
Rise = 6% * 2.5 miles * 5280 feet/mile = 0.06 * 2.5 * 5280 feet = 792 feet
Therefore, a highway with a six percent grade would rise 792 feet in 2.5 miles of horizontal distance.
b. If we want to calculate the rise in feet for a longer horizontal distance, like 90 miles, with a constant percent grade, we can use the same formula:
Rise (in feet) = Percent Grade (%) * Horizontal Distance (in miles) * 5280 (feet per mile)
For a highway with a six percent grade and a horizontal distance of 90 miles:
Rise = 6% * 90 miles * 5280 feet/mile = 0.06 * 90 * 5280 feet = 28,512 feet
Therefore, a highway with a six percent grade would rise 28,512 feet in 90 miles of horizontal distance.
c. The relationship between percent grade and slope is as follows:
Slope = Rise / Horizontal Distance
Percent Grade = (Rise / Horizontal Distance) * 100
In other words, slope is the ratio of the rise to the horizontal distance, and percent grade is the slope expressed as a percentage.
For example, if a highway rises 100 feet over a horizontal distance of 500 feet, the slope is:
Slope = 100 feet / 500 feet = 0.2
The percent grade is:
Percent Grade = 0.2 * 100 = 20%
So, the percent grade is a way to express the slope as a percentage.