Kim and Pam are participating in a two-minute partner competition. It takes Kim 6 seconds to solve each puzzle and it takes Pam 3 seconds to hit a target with a ball. They cannot solve puzzles and hit targets at the same time.

Question

Which of the following graphs represents this situation?

Answer 1

To represent the situation with Kim and Pam, we need to consider how many puzzles Kim can solve and how many targets Pam can hit in 2 minutes (120 seconds).

1. Kim's Puzzle Solving:

   • It takes Kim 6 seconds to solve each puzzle.
   • In 120 seconds, the number of puzzles Kim can solve is: \[ \frac{120 \text{ seconds}}{6 \text{ seconds per puzzle}} = 20 \text{ puzzles} \]

2. Pam's Target Hitting:

   • It takes Pam 3 seconds to hit each target.
   • In 120 seconds, the number of targets Pam can hit is: \[ \frac{120 \text{ seconds}}{3 \text{ seconds per target}} = 40 \text{ targets} \]

3. Graph Representation:

   • On the x-axis, we can represent the number of puzzles solved by Kim (0 to 20).
   • On the y-axis, we can represent the number of targets hit by Pam (0 to 40).
   • If we create a scenario where one partner is maximizing their time while the other is doing the same, we can expect a linear relationship where:
     • If Kim spends all her time solving puzzles, she reaches the maximum of 20 puzzles and 0 targets.
     • Conversely, if Pam spends all her time hitting targets, she reaches the maximum of 0 puzzles and 40 targets.

The resulting graph represents a line running from (20, 0) to (0, 40).

The graph must demonstrate that as one partner does more of their task (Kim solving more puzzles), the other partner does less of theirs (Pam hitting fewer targets), illustrating a trade-off in the competition where one’s gain is the other’s loss.

Thus, the graph representing this situation would be a straight line between the points (20, 0) and (0, 40).

Answer 2

is the line dashed or not

Answer 3

The line representing the relationship between the number of puzzles Kim can solve and the number of targets Pam can hit can vary based on specific instruction, but generally:

• If the graph indicates potential outcomes when both partners are working simultaneously (as in a competition scenario where they could be maximizing their time but still cannot perform both actions at once), the line is typically solid. This solid line represents all possible combinations of the tasks they can complete under the time constraint.

• If the scenario implies that there are specific conditions under which they cannot trade off their tasks (like if one person could never solve puzzles while the other is hitting targets), a dashed line might indicate that certain combinations are not possible.

In this case, since they are trying to optimize their own tasks simultaneously, the most appropriate representation is a solid line between (20, 0) and (0, 40), signifying the possible combinations of tasks they can complete in the available time. Therefore, the line should be solid.

Answer 4

wheres the shaded area

Answer 5

In the context of the problem with Kim and Pam, the shaded area on the graph represents the feasible combinations of puzzles solved by Kim and targets hit by Pam within the given time constraint of 120 seconds.

Feasible Region

Since the maximum number of puzzles Kim can solve is 20 and the maximum number of targets Pam can hit is 40, the feasible region is bounded by the lines:

• The vertical line at \( x = 20 \) (maximum puzzles solved by Kim).
• The horizontal line at \( y = 40 \) (maximum targets hit by Pam).
• The line connecting the points \( (20, 0) \) and \( (0, 40) \).

Characteristics of the Shaded Area

1. Vertices:

   • (0, 0): No time spent on either task.
   • (20, 0): All time spent by Kim solving puzzles, and none by Pam shooting targets.
   • (0, 40): All time spent by Pam hitting targets, and none by Kim solving puzzles.

2. Bounded Area:

   • The region below and to the left of the line segment from \( (20, 0) \) to \( (0, 40) \) would be shaded, indicating all the combinations of puzzles solved and targets hit that are possible within the 120 seconds.

3. Area Explanation:

   • The shaded area represents the scenarios in which both Kim and Pam are maximizing their efforts. It includes all combinations from doing no tasks to the extremes of either completing all puzzles or hitting all targets.

Conclusion

In summary, the shaded area on the graph should be the region under the straight line segment connecting \( (20, 0) \) and \( (0, 40) \), including the x-axis and y-axis to the left and down from the vertices. This shaded area illustrates the feasible combinations of tasks they can both accomplish within their two-minute timeframe.