Question

dan is competing in the sponge - soak relay race during field day. he runs across the field to a bucket, fills up a sponge with water from the bucket, and then runs back to tag the next racer. which graph could show dans distance from the bucket over time? two graphs are shown with y - axis labeled distance from bucket (ft.) and x - axis labeled time (sec.). the first graph has a line starting at a positive y - value, decreasing to 0, then increasing back to the original y - value. the second graph has a line starting at 0, increasing to a positive y - value, staying constant for a while, then decreasing back to 0.

Explanation:

Step1: Analyze Dan's motion phases

Dan starts at some distance from the bucket (initial distance > 0), runs towards the bucket (distance decreases to 0), stops to fill the sponge (distance remains 0 for a short time), then runs back (distance increases back to the initial distance). Wait, no—wait, the problem says "runs across the field to a bucket, fills up a sponge with water from the bucket, and then runs back to tag the next racer." Wait, maybe I misread. Wait, when he runs to the bucket: his distance from the bucket starts at some value (let's say he starts at his team's position, distance D from bucket), then runs to bucket: distance decreases to 0. Then he fills the sponge: so during filling, time passes but distance from bucket is 0 (so the graph should have a horizontal line at y=0 for the time he's filling). Then he runs back: distance increases from 0 back to D. Wait, but the first graph: left graph: starts at y>0, decreases to 0, then has a horizontal line at 0 (wait, no, the left graph: after decreasing to 0, there's a horizontal segment? Wait, the left graph: the first line goes down to 0, then a horizontal line (time passes, distance 0), then a line going up. Wait, the right graph: starts at 0, increases to a max, horizontal, then decreases to 0. But Dan starts at a distance from the bucket (not at 0), so initial distance is positive. So let's re-examine:

1. Initial state: Dan is at his starting point, distance from bucket is some positive value (y-intercept > 0).
2. Phase 1: Runs towards bucket. Distance from bucket decreases linearly (line with negative slope) until he reaches the bucket (distance = 0).
3. Phase 2: Fills the sponge. Time passes, but he's at the bucket, so distance from bucket remains 0 (horizontal line, y=0) for the duration of filling.
4. Phase 3: Runs back to tag the next racer. Distance from bucket increases linearly (line with positive slope) back to the initial distance (since he returns to the starting point? Wait, no—"runs back to tag the next racer"—so he goes back to his team's position, which is the initial distance from the bucket. So distance increases from 0 to initial distance.

Now look at the graphs:

- Left graph: y-intercept > 0 (starts at positive distance), first line down to 0 (phase 1), then horizontal (phase 2, filling), then line up (phase 3, running back).
- Right graph: starts at 0, goes up (so initial distance 0, which is wrong, because he starts away from bucket), then horizontal, then down. So right graph is wrong because initial distance should be positive.

Wait, maybe I misread the problem. Wait, the problem says "Dan is competing in the sponge-soak relay race... He runs across the field to a bucket, fills up a sponge with water from the bucket, and then runs back to tag the next racer." So "runs across the field to a bucket"—so he starts at his team's area, distance from bucket is D. Then runs to bucket: distance decreases to 0. Then fills sponge: time passes, distance 0. Then runs back: distance increases to D (since he returns to tag the next racer, so back to starting point). So the graph should have:

- Initial y (distance) > 0.
- Decrease to 0 (slope negative).
- Horizontal at 0 (time for filling).
- Increase back to initial y (slope positive).

Looking at the left graph: the first line is decreasing (distance to bucket decreases), then a horizontal segment (distance 0, filling), then increasing (distance back). The right graph starts at 0, so initial distance 0, which is incorrect. So the left graph (first graph) is correct? Wait, but the left graph: after decreasing to 0, there's a horizontal line (time passes, distance 0), then increasing. Wait, the left graph's x-axis: the first line goes down to 0, then a horizontal line (so time when distance is 0), then a line going up. So that matches: start at distance >0, run to bucket (distance 0), wait (distance 0), run back (distance increases). The right graph starts at 0, so he starts at the bucket, which is wrong. So the correct graph is the left one (the first graph shown).

Wait, but the problem's graphs: let's check the axes. Left graph: y-axis is distance from bucket (ft), x-axis time (sec). Initial y is positive (starts at distance from bucket), then decreases to 0 (reaches bucket), then horizontal (filling), then increases back. Right graph: initial y=0 (starts at bucket), then increases (runs away), then horizontal (stops), then decreases (runs back to bucket). But Dan starts away from the bucket, so initial distance is positive. So left graph is correct.

Step2: Eliminate the wrong graph

The right graph starts at distance 0 (Dan starts at the bucket), which is incorrect because he runs to the bucket (so he starts away from it). The left graph starts at positive distance, decreases to 0 (reaches bucket), has a horizontal segment (filling), then increases back (runs back), which matches the motion: start away, run to bucket, fill, run back.

Answer:

The left graph (the first one, with initial distance >0, decreasing to 0, horizontal at 0, then increasing) is the correct one. (Assuming the left graph is the first of the two, with the y-intercept positive, a line down to 0, a horizontal line, then a line up.)