Question

Rachel hikes at a steady rate from a ranger station to a campground that is 20 mi away. After 2 h, she is 13 mi from the campground. After 4 h, she is 6 mi from the campground. A graph shows her distance from the campground y, in miles, after x hours. What is the slope of the graph, and what does it represent?A.20; Rachel's initial distance from the campgroundB.−3.5; the rate at which Rachel's distance from the campground changes per hourC.3.5; Rachel's initial distance from the campgroundD.20; Rachel's final distance from the campground

Answer 1

Given the information in the question, the following steps help to solve the question

Step 1: Write out the mathematical interpretations of the statements:

After 2 hours, Rachel is 13 miles from the campground.

After 4 hours, Rachel is 6 miles from the campground.

Step 2: Write the interpretations in form of points, we have:

x as the number of hours and y as the number of miles from the campground.

Therefore, we have two points (2,13) and (4,6).

`(x_1,y_1_{})(x_2,y)=(2,13)and(4,6)`

Step 3: We find the slope of the graph using the point gotten:

The equation of the line passing through the points is:

`\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1_{}}(x-x_(1)) \\ y-13=(6-13)/(4-2)(x-2) \\ y-13=(-7)/(2)(x-2) \\ y-13=-3.5(x-2) \\ y-13=-3.5x+7 \\ y=-3.5x+7+13 \\ y=-3.5x+20 \end{gathered}`

Step 4: Get the slope using the general equation of a line

`\begin{gathered} y=bx+c \\ \text{where b is the slope and c is the y-intercept} \\ \text{if }y=-3.5x+20 \\ \text{then, slope(b)=-3.5} \end{gathered}`

Hence, the slope of the graph is -3.5 and this means that the rate at which Rachel's distance from the campground changes per hour is 3.5 miles which is option B.