Question

1. a group of hikers is progressing steadily along an uphill trail. the graph shows their elevation (or height above sea level), in feet, at each distance from the start of the trail, in miles.

a. what is the slope of the graph? show your reasoning.
b. what does the slope tell us about this situation?
c. write an equation that represents the relationship between the hikers distance from the start of the trail, x, and their elevation, y.
d. does the equation y - 250x = 500 represent the same relationship between the distance from the start of the trail and the elevation? explain your reasoning.

Explanation:

Step1: Find the slope

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points from the graph: \((0,500)\) and \((2,1000)\).
\(m=\frac{1000 - 500}{2-0}=\frac{500}{2}=250\)

Step2: Interpret the slope

The slope \(m = 250\) means that for every 1 - mile increase in the distance \(x\) from the start of the trail, the elevation \(y\) increases by 250 feet.

Step3: Write the equation

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. From the graph, when \(x = 0\), \(y=500\) (so \(b = 500\)) and \(m = 250\).
The equation is \(y=250x + 500\).

Step4: Check the start of the trail

When \(x = 0\) (start of the trail), substitute into \(y-250x=500\).
\(y-250\times0=500\), so \(y = 500\).

Answer:

a. The slope of the graph is \(250\). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with points \((0,500)\) and \((2,1000)\), \(m=\frac{1000 - 500}{2-0}=250\). It means the elevation increases by 250 feet for every 1 - mile increase in distance from the start of the trail.
b. The slope \(250\) tells us the rate of change of elevation with respect to distance. For each mile walked along the trail, the elevation goes up by 250 feet.
c. The equation is \(y = 250x+500\) (derived from \(y=mx + b\) where \(m = 250\) and \(b = 500\) (the \(y\) - intercept, elevation at \(x = 0\))).
d. Yes. When \(x = 0\) (start of the trail), in the equation \(y-250x=500\), we get \(y=500\) (elevation at the start).