Question

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. When \(x = 0\), \(y=500\) (the \(y -\)intercept) and when \(x = 8\), \(y=250\times8 + 500=2500\). Using the formula with \((x_1,y_1)=(0,500)\) and \((x_2,y_2)=(8,2500)\), \(m=\frac{2500 - 500}{8-0}=\frac{2000}{8}=250\).

Step2: Interpret the slope

The slope \(m = 250\) means that for every 1 - mile increase in the distance along the trail (\(x\)), 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. Here, \(m = 250\) and \(b = 500\), so the equation is \(y=250x+500\).

Step4: Check the given equation

Given \(y - 250x=500\), we can rewrite it in slope - intercept form \(y=250x + 500\). When \(x = 0\) (start of the trail), \(y=500\) (elevation at the start).

Answer:

a. The slope of the graph is \(250\). It means that for each mile along the trail, the elevation increases by \(250\) feet.
b. The slope \(250\) tells us the rate of change of elevation with respect to the distance along the trail. That is, for every 1 - mile increase in distance, the elevation increases by \(250\) feet.
c. The equation is \(y = 250x+500\) (derived from slope - intercept form \(y=mx + b\) with \(m = 250\) and \(b = 500\)).
d. When \(x = 0\) (start of the trail) in the equation \(y-250x = 500\) (or \(y=250x + 500\)), \(y=500\). So the elevation at the start of the trail is \(500\) feet.