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.