A highway is constructed up a mountain pass at a constant incline. At a distance of 3.6 mi… — ACT

A highway is constructed up a mountain pass at a constant incline. At a distance of $3.6$ miles from the starting toll gate, the elevation of the highway is $1,420$ feet above sea level. At a distance of $8.4$ miles from the toll gate, the elevation is $3,100$ feet above sea level. What is the slope of the highway's elevation profile, in feet per mile?

Answer: 350 feet per mile

Answer

The slope of the highway's elevation profile is 350 feet per mile.

The slope represents the constant rate of change of elevation per mile. By defining the coordinates as

(x_1, y_1) = (3.6, 1420)

and

(x_2, y_2) = (8.4, 3100)

, we calculate the slope using the slope formula. Substituting the values gives

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8} = 350

Step-by-Step Solution

Represent the given data as coordinate points where the x-coordinate represents the distance in miles and the y-coordinate represents the elevation in feet.

The coordinate points are

(3.6, 1420)

and

(8.4, 3100)

Slope is the change in the dependent variable (elevation) per unit change in the independent variable (distance).

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to find the rate of change.

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8}

The slope represents the constant rate of change of elevation relative to the distance traveled.

Perform the division to find the numerical slope.

350

Dividing 1680 by 4.8 yields the exact value of 350.

Key Concept

The slope of a line represents a constant rate of change between two variables, calculated as the change in the vertical coordinate divided by the change in the horizontal coordinate.

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