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Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt

tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be
at least one adult and at least one child at the party. She has budgeted $300 for the party and that
is the maximum amount she wants to spend.

What are the constraints of the situation? Describe them.

3 answers

To find the constraints of the situation described, we need to identify the conditions based on the costs and requirements for the party. Here are the constraints:

  1. Total Cost Constraint: The total cost of the party must not exceed the budgeted amount of $300. \[ 150 + 10.50c + 5.50a \leq 300 \] Here, \(c\) represents the number of children and \(a\) represents the number of adults.

  2. Minimum Children Constraint: There must be at least one child at the party. \[ c \geq 1 \]

  3. Minimum Adults Constraint: There must be at least one adult at the party. \[ a \geq 1 \]

  4. Non-negativity Constraints: The number of children and adults cannot be negative.

    • While implied by the minimum constraints, stated explicitly, we have: \[ c \geq 0 \] \[ a \geq 0 \]

To summarize the constraints mathematically:

  • Budget constraint: \(150 + 10.50c + 5.50a \leq 300\)
  • Minimum children: \(c \geq 1\)
  • Minimum adults: \(a \geq 1\)

These constraints define the allowable combinations of children and adults that Meha can invite to the party while staying within budget and satisfying minimum attendance requirements.

Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt
tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be
at least one adult and at least one child at the party. She has budgeted $300 for the party and that
is the maximum amount she wants to spend.

Give a recommendation to Meha as to how many adults and children you think she should
invite to the party. Justify your recommendation.

To provide a recommendation on how many adults and children Meha should invite to the party, we need to analyze the constraints and maximize the number of guests within the budget of $300.

1. Establishing the Budget Equation:

Given the costs:

  • Flat cost: $150
  • Cost per child: $10.50
  • Cost per adult: $5.50

The total cost \(C\) can be expressed as: \[ C = 150 + 10.50c + 5.50a \]

2. Setting Up the Budget Constraint: Since Meha's aunt has a budget of $300: \[ 150 + 10.50c + 5.50a \leq 300 \]

Simplifying this gives: \[ 10.50c + 5.50a \leq 150 \]

3. Considering Minimum Requirements: The problem states that at least one child and one adult must be present: \[ c \geq 1 \] \[ a \geq 1 \]

4. Exploring Possible Combinations: To find the optimal number of guests, we can experiment with values for \(c\) and \(a\). We will check a couple of scenarios while keeping the budget in mind:

  • Scenario 1 (1 adult, 13 children): \[ C = 150 + 10.50(13) + 5.50(1) = 150 + 136.5 + 5.5 = 292 \] (Total guests: 1 adult + 13 children = 14 guests)

  • Scenario 2 (2 adults, 10 children): \[ C = 150 + 10.50(10) + 5.50(2) = 150 + 105 + 11 = 266 \] (Total guests: 2 adults + 10 children = 12 guests)

  • Scenario 3 (3 adults, 7 children): \[ C = 150 + 10.50(7) + 5.50(3) = 150 + 73.5 + 16.5 = 240 \] (Total guests: 3 adults + 7 children = 10 guests)

  • Scenario 4 (4 adults, 4 children): \[ C = 150 + 10.50(4) + 5.50(4) = 150 + 42 + 22 = 214 \] (Total guests: 4 adults + 4 children = 8 guests)

  • Scenario 5 (5 adults, 3 children): \[ C = 150 + 10.50(3) + 5.50(5) = 150 + 31.5 + 27.5 = 209 \] (Total guests: 5 adults + 3 children = 8 guests)

Recommendation: Based on the calculations, Scenario 1 with 1 adult and 13 children is the best option with 14 total guests while still being well within the budget of $300 (total cost is $292).

Justification: This scenario maximizes the number of children invited, which is often a priority for a birthday party. Involving 1 adult provides the necessary supervision while allowing for a lively atmosphere with many children, ensuring a fun celebration for the cousin. Additionally, any extra funds left in the budget could potentially be used for treats or small party favors, making the event even more enjoyable.