Question: STRUCTURE Jazmin is participating in a 25.5-kilometer charity walk. She walks at a rate of 4.25 km per hour. Jazmin walks at the same pace for the entire event.
a. Write an equation in slope-intercept form for the remaining distance, y, in kilometers of walking for x hours.
y = 4.25x + 25.5
b. Graph the function on a separate sheet of paper.
c. What do the x- and y-intercepts represent in this situation?
The x-intercept is 6 and represents the number of hours it will take Jazmin to complete the walk.
The y-intercept is 25.5 and represents the length of the walk.
Part B
d. Determine the domain and range of the function and interpret their meaning in the context of the situation.
Time cannot be negative and Jazmin completes the walk in 6 hours, so the domain is {x | 0 64 x 64 6}. Distance cannot be negative and the length of the walk is 25.5 kilometers, so the range is {y | 0 64 y 64 25.5}
e. After Jazmin has walked 17 kilometers, how much longer will it take her to complete the walk? Explain how you can use your graph to answer the question.
Using the graph, I can determine the value of x when y equals 17 km, and use the value of the x-intercept. The value of x is 2 when y = 8.5 and the x-intercept is 6. Therefore, Jazmin has 4 hours more to walk.
top-left: a small graph would show a line with positive slope 4.25, crossing the y-axis at 25.5 and the x-axis at 6.
1. **State the problem:** Jazmin is walking a total distance of 25.5 km at a rate of 4.25 km/h. We want to write an equation for the remaining distance $y$ after $x$ hours, analyze intercepts, domain, range, and find how much longer she will walk after 17 km.
2. **Write the equation:** The problem states $y = 4.25x + 25.5$, but this does not represent remaining distance because distance should decrease as time increases. The correct equation for remaining distance $y$ after $x$ hours is:
$$y = 25.5 - 4.25x$$
This is because Jazmin starts with 25.5 km to walk and reduces distance by $4.25$ km each hour.
3. **Interpret intercepts:**
- The $y$-intercept is when $x=0$, so $y=25.5$. This means at the start, the remaining distance is 25.5 km.
- The $x$-intercept is when $y=0$, solve:
$$0 = 25.5 - 4.25x$$
$$4.25x = 25.5$$
$$x = \frac{25.5}{4.25} = 6$$
This means Jazmin finishes the walk after 6 hours.
4. **Domain and range:**
- Domain: Time $x$ cannot be negative and cannot exceed 6 hours, so
$$0 \leq x \leq 6$$
- Range: Remaining distance $y$ cannot be negative and starts at 25.5 km, so
$$0 \leq y \leq 25.5$$
5. **After walking 17 km, find remaining time:**
- Remaining distance after 17 km is:
$$y = 25.5 - 17 = 8.5$$
- Use the equation to find $x$ when $y=8.5$:
$$8.5 = 25.5 - 4.25x$$
$$4.25x = 25.5 - 8.5 = 17$$
$$x = \frac{17}{4.25} = 4$$
- Jazmin has walked for 4 hours and since total time is 6 hours, she has:
$$6 - 4 = 2$$
hours left to walk.
6. **Summary:**
- Correct equation: $$y = 25.5 - 4.25x$$
- $y$-intercept: 25.5 km (start distance)
- $x$-intercept: 6 hours (time to finish)
- Domain: $0 \leq x \leq 6$
- Range: $0 \leq y \leq 25.5$
- After 17 km, Jazmin has 2 hours left to walk.