Seth, Summer, and Loriece are hiking on the same trail in North Georgia. Summer started her hike early in the morning and Seth began his hike later in the day. Seth’s hike can be modeled by the equation y = ⅔ x , where x represents the hours spent hiking and y is the distance hiked in miles.

Question

When Seth began his hike, Summer was already 5 miles from the beginning of the trail. Two hours after Seth began his hike, Summer was 8 miles from the beginning of the trail.
Another hiker, Loriece, is hiking on the same trail. She began her hike at the other end of the trail. When Seth begins his hike, Loriece also begins 9 miles away. After Seth has hiked for 3 hours, Loriece is 7 miles from where he began.
Can you graph this scenario?

Answer 1

To graph the scenario involving Seth, Summer, and Loriece, we need to establish a coordinate system where the x-axis represents time in hours and the y-axis represents distance from the start of the trail in miles.

Step 1: Set Up the Coordinate System

• The vertical axis (y-axis) will represent the distance in miles from the start of the trail.
• The horizontal axis (x-axis) will represent the time in hours.

Step 2: Model Seth's Hike

The equation given for Seth is:

• \( y = \frac{2}{3}x \)

This means that for every hour Seth hikes, he covers \(\frac{2}{3}\) of a mile. Since Seth begins his hike later, we'll assume he starts at \(x = 0\).

• At \(x = 0\) (when Seth begins his hike), \(y = 0\) miles.
• At \(x = 1\): \(y = \frac{2}{3}(1) = \frac{2}{3}\) miles
• At \(x = 2\): \(y = \frac{2}{3}(2) = \frac{4}{3}\) miles
• At \(x = 3\): \(y = \frac{2}{3}(3) = 2\) miles
• At \(x = 4\): \(y = \frac{2}{3}(4) = \frac{8}{3}\) miles
• At \(x = 5\): \(y = \frac{2}{3}(5) = \frac{10}{3}\) miles
• At \(x = 6\): \(y = \frac{2}{3}(6) = 4\) miles

Step 3: Model Summer's Hike

Summer started 5 miles ahead when Seth began, and two hours later, she is 8 miles from the beginning.

• When Seth starts \((x = 0)\), Summer is 5 miles from the start, so we can use a point \((0, 5)\).
• Two hours later, Seth has been hiking for 2 hours, during which time Summer has moved from 5 miles to 8 miles, adding 3 miles in 2 hours. Therefore, her rate is \(\frac{3 \text{ miles}}{2 \text{ hours}} = 1.5 \text{ miles per hour}\).

The equation for Summer’s distance as a function of time since Seth started can be modeled as:

• \( y = 1.5x + 5 \)

Step 4: Model Loriece's Hike

Loriece begins her hike 9 miles away from where Seth starts. After 3 hours of Seth hiking, she is 7 miles from the start, meaning she has covered:

• \(9 - 7 = 2\) miles.

Since this takes 3 hours, her rate is: \(\frac{2 \text{ miles}}{3 \text{ hours}} = \frac{2}{3} \text{ miles per hour}\).

So, at the start of Seth's hike (when \( x = 0 \)), Loriece is at \(y = 9\). The equation for Loriece’s distance as a function of time since Seth started can be modeled as:

• \(y = 9 - \frac{2}{3}x\)

Step 5: Plotting the Graph

1. Seth's equation: \(y = \frac{2}{3}x\)

   • Plot points from \(x = 0\) to your desired \(x\), continuing upward.

2. Summer's equation: \(y = 1.5x + 5\)

   • Starts at \( (0, 5) \) and increases at \(1.5\) miles/hour.

3. Loriece's equation: \(y = 9 - \frac{2}{3}x\)

   • Starts at \( (0, 9) \) and decreases at \(\frac{2}{3}\) miles/hour.

Graphing the Lines

Using graph paper or a graphing software:

• For Seth: Start at the origin and draw a line with a slope of \(\frac{2}{3}\).
• For Summer: Start at the point (0, 5) and draw a line with a slope of \(1.5\).
• For Loriece: Start at the point (0, 9) and draw a line with a slope of \(-\frac{2}{3}\).

Label the lines and points at significant positions, and you will visually represent the scenario!

This graph will show how the distances of each hiker change over time, illustrating their hiking paths along the trail in North Georgia.