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 ${x | 0 \leq x \leq 6}$. Distance cannot be negative and the length of the walk is 25.5 kilometers, so the range is ${y | 0 \leq y \leq 25.5}$.
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 8.5 km, and use the value of the $x$-intercept. The value of $x$ is 4 when $y = 8.5$ and the $x$-intercept is 6. Therefore, Jazmin has 2 hours more to walk.
1. **State the problem:** Jazmin walks a 25.5 km charity walk at 4.25 km/h. We want to find the equation for the remaining distance $y$ after $x$ hours, interpret intercepts, domain, range, and find time left after walking 17 km.
2. **Write the equation for remaining distance:**
Jazmin walks at 4.25 km/h, so distance walked after $x$ hours is $4.25x$.
Remaining distance $y$ is total distance minus distance walked:
$$y = 25.5 - 4.25x$$
This is the correct slope-intercept form where slope is $-4.25$ and $y$-intercept is 25.5.
3. **Check intercepts:**
- $y$-intercept: When $x=0$, $y=25.5$. This means at start, remaining distance is full 25.5 km.
- $x$-intercept: Set $y=0$ to find when Jazmin finishes:
$$0 = 25.5 - 4.25x$$
$$4.25x = 25.5$$
$$x = \frac{25.5}{4.25} = 6$$
So, it takes 6 hours to complete the walk.
4. **Domain and range:**
- Domain: Time $x$ cannot be negative and cannot exceed 6 hours (completion time), 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. **Find time left after walking 17 km:**
Remaining distance after walking 17 km is:
$$y = 25.5 - 17 = 8.5$$
Use equation to find $x$ when $y=8.5$:
$$8.5 = 25.5 - 4.25x$$
$$4.25x = 25.5 - 8.5 = 17$$
$$x = \frac{17}{4.25} = 4$$
Since total time is 6 hours, time left is:
$$6 - 4 = 2 \text{ hours}$$
Jazmin has 2 hours more to walk.
**Final answers:**
- Equation: $$y = 25.5 - 4.25x$$
- $x$-intercept: 6 hours (time to finish)
- $y$-intercept: 25.5 km (total distance)
- Domain: $$0 \leq x \leq 6$$
- Range: $$0 \leq y \leq 25.5$$
- Time left after 17 km: 2 hours