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B. Is it possible for the party to include 10 children and five adults? Use your inequality from

part A to check. Explain your answer.

5 answers

To determine if it's possible to include 10 children and 5 adults in a party, you would typically have an inequality that represents the maximum number of children and adults that can be included based on certain constraints (like space or resources).

Assuming the inequality from part A is something like:

  • Let \( c \) represent the number of children.
  • Let \( a \) represent the number of adults.
  • Suppose the inequality was \( c + 2a \leq 30 \) (for example, where 2 adults count as the same resource usage as 1 child, or whatever the specific constraint may be).

Now, plug in the numbers from your scenario:

  1. \( c = 10 \) (children)
  2. \( a = 5 \) (adults)

Now substitute into the inequality:

\[ 10 + 2(5) \leq 30 \]

Calculating this gives:

\[ 10 + 10 \leq 30 \] \[ 20 \leq 30 \] (This is true)

Since the inequality holds true, it is possible to include 10 children and 5 adults at the party.

Summary

  1. Check the inequality by plugging in the values for children and adults.
  2. If the inequality holds true, then it is feasible to have that combination of children and adults at the party. In this case, having 10 children and 5 adults is possible based on the example inequality provided.
What are the constraints of the situation? Describe them.

The constraints of a situation like hosting a party with a specific number of children and adults typically include several factors that can limit the number of attendees or how they can participate. Here are some common constraints that could apply:

  1. Space Limitations:

    • There might be a maximum number of people that can comfortably fit in the venue (e.g., a house, backyard, or rented space). For example, you might have a total capacity limit of 30 people.

  2. Budgetary Constraints:

    • The total cost of hosting the party (including food, drinks, decorations, activities, and possibly venue rental) may limit the number of attendees. Each adult might cost more in terms of food and resources compared to children.

  3. Resource Availability:

    • You might have a limited amount of supplies such as seating, food, drinks, and party favors. For example, you might only have enough chairs for 20 people or enough food for a certain number of individuals.

  4. Adult Supervision Requirements:

    • Depending on the age of the children, there may be regulations or safety considerations regarding the adult-to-child ratio. For example, you might need at least one adult for every 5 children to ensure adequate supervision.

  5. Time Constraints:

    • The duration of the party could limit how long you can accommodate certain activities, which may indirectly affect how many people can attend.

  6. Health and Safety Regulations:

    • Depending on local regulations (especially in light of past health concerns like the COVID-19 pandemic), restrictions might be in place regarding the number of attendees, social distancing, or sanitation practices.

  7. Activity Limitations:

    • If the party includes specific activities (e.g., games, crafts, or food preparation), there could be limits on how many people can participate in those activities at the same time.

  8. Time of Day and Day of the Week:

    • Certain times might be more suitable for families with children, affecting availability and participation, depending on school and work schedules.

By understanding these constraints, you can better plan the party to accommodate both the desired number of children and adults while ensuring a successful and enjoyable event for everyone involved.

E. Show all the viable solutions to the problem as points on the graph.

To visualize the viable solutions to the problem involving the number of children and adults at a party, we first need to establish the inequality constraints based on the scenario.

Let's assume we have two constraints:

  1. \( c + 2a \leq 30 \) (an example constraint where children count as 1 unit and adults count as 2 units)
  2. \( c \leq 20 \) (a maximum number of children)
  3. \( a \leq 15 \) (a maximum number of adults)

Where:

  • \( c \) is the number of children.
  • \( a \) is the number of adults.

Steps to Graph the Solutions

  1. Set Up the Axes:

    • Draw a coordinate system where the x-axis represents the number of children (\( c \)) and the y-axis represents the number of adults (\( a \)).

  2. Graph Each Constraint:

    • From \( c + 2a \leq 30 \):

      • Rearranging gives \( a \leq \frac{30 - c}{2} \).
      • When \( c = 0 \), \( a = 15 \) (point: \( (0, 15) \)).
      • When \( c = 30 \), \( a = 0 \) (point: \( (30, 0) \)).
      • The line connecting these points is bordered by the area below it.

    • From \( c \leq 20 \):

      • Draw a vertical line at \( c = 20 \).

    • From \( a \leq 15 \):

      • Draw a horizontal line at \( a = 15 \).

  3. Shading the Regions:

    • Shade the area that satisfies all inequalities, which is below the lines representing the constraints.

  4. Viable Integer Solutions:

    • The viable integer solutions are the points within the shaded area, including points along the edges of the lines but inside the inequalities.

Example Points

Let’s evaluate some integer points that fall within these constraints:

  • \( (0, 0) \) — 0 children, 0 adults
  • \( (10, 5) \) — 10 children, 5 adults
  • \( (20, 0) \) — 20 children, 0 adults
  • \( (20, 10) \) — 20 children, 10 adults
  • \( (15, 7.5) \) but since we require integers, use \( (15, 7) \) or \( (15, 8) \)
  • \( (0, 15) \) — 0 children, 15 adults

Conclusion

The viable points are those that satisfy all of the constraints, and you can plot these on the graph. The exact points will depend on the constraints you have established. If every point within the shaded area retains non-negative integer values, each will represent a viable solution to the party's constraints.

If you have specific constraints or a range of values in mind, let me know, and I can help narrow down the points further or create a more specific graphical representation!