The following graph shows the position functions of two runners, A and B, who run a 100-meter race and finish in a tie.

On the coordinate plane the horizontal axis is labeled t (seconds) and the vertical axis is labeled s (meters). A line and a curve are graphed.
The line labeled A begins at the origin, goes up and right, and ends at the point (14, 100).
The curve labeled B begins at the origin, goes up and right becoming more steep, passes through the approximate point (9.5, 55) where the curve B is furthest from the line A, and ends at the point (14, 100).
(a)
Describe and compare how the runners run the race.

Runner A runs the entire race at the same velocity. Runner B starts the race at a faster velocity than runner A but finishes the race at a slower velocity.
Runner A starts the race at a slower velocity than runner B but finishes the race at a faster velocity. Runner B runs the entire race at the same velocity.    
Runner A starts the race at a faster velocity than runner B but finishes the race at a slower velocity. Runner B runs the entire race at the same velocity.
Runner A and runner B both start and end the race a the same velocities.
Runner A runs the entire race at the same velocity. Runner B starts the race at a slower velocity than runner A but finishes the race at a faster velocity.
Correct: Your answer is correct.
(b)
How long after the race began (in seconds) is the distance between the runners the greatest?
9.5
 
Correct: Your answer is correct.
 s
(c)
How long after the race began (in seconds) do the runners have the same velocity?
7
 
Incorrect: Your answer is incorrect.
 s

Let's analyze the problem step by step to understand why part (c) might be incorrect.

### Given Information:
- **Runner A** has a **constant velocity** throughout the race, as indicated by the straight line for A’s position function.
- **Runner B** has a **changing velocity**, as indicated by the curved graph for B’s position function.

### Part (c) asks: "How long after the race began do the runners have the same velocity?"

#### Key Insights:
- **Runner A**: Since A's position is represented by a straight line, the velocity is constant throughout the race. We can calculate A's velocity as the slope of the line:
  \[
  \text{Velocity of A} = \frac{\Delta s}{\Delta t} = \frac{100 \, \text{meters}}{14 \, \text{seconds}} = \frac{100}{14} \approx 7.14 \, \text{m/s}.
  \]
  
- **Runner B**: B's velocity changes because the graph is curved. The velocity at any time is the slope of the tangent line to the curve at that point. So, to find when A and B have the same velocity, we need to find when the slope of B's curve equals A's constant velocity (7.14 m/s).

#### Correct Approach:
1. Since B starts faster than A but slows down over time, there is a point where B’s velocity decreases to match A’s constant velocity.
2. To find this, we need to estimate the slope of the curve for B and determine when that slope is approximately equal to 7.14 m/s.

Looking at the graph, the steepest part of B’s curve happens early in the race (around 0 to 7 seconds). As time progresses, B's velocity decreases and approaches A’s constant velocity. From the problem, the incorrect answer you submitted was 7 seconds, but based on typical behavior in such curves, it’s likely the runners have the same velocity earlier, possibly between 5 to 6 seconds.

Thus, the correct answer is likely **5 seconds** based on the general shape of the graph and how velocities change in such scenarios.

### Clarifications and Next Steps:
Would you like me to explain further details about how to interpret the tangent slope or the graph? Feel free to ask if you have any questions or need additional context!

Here are five questions to deepen understanding:
1. How do you determine velocity from a position-time graph?
2. Why is the slope of a straight line constant but the slope of a curve variable?
3. How does the steepness of a curve relate to the speed of the runner?
4. What methods can we use to estimate the slope of a curve at a given point?
5. How can you compare the velocities of two runners from their position-time graphs?

**Tip**: In calculus, the velocity is the derivative of the position function with respect to time. Understanding this concept will help in analyzing curved graphs more precisely.

The following graph shows the position functions of two runners, A and B, who run a 100-meter race and finish in a tie. On the coordinate plane the horizontal axis is labeled t (seconds) and the vertical axis is labeled s (meters). A line and a curve are graphed. The line labeled A begins at the origin, goes up and right, and ends at the point (14, 100). The curve labeled B begins at the origin, goes up and right becoming more steep, passes through the approximate point (9.5, 55) where the curve B is furthest from the line A, and ends at the point (14, 100). (a) Describe and compare how the runners run the race. (b) How long after the race began (in seconds) is the distance between the runners the greatest? (c) How long after the race began (in seconds) do the runners have the same velocity?

This problem explores how to analyze the velocities of two runners using position-time graphs. Runner A maintains constant velocity, while Runner B's velocity changes over time. Learn how to calculate velocity, interpret the slope of a curve, and understand when the runners' velocities match. Suitable for students in Grades 9-12.

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