Answer :
Alright, let's analyze the given information step-by-step and verify each of Gale's claims.
### Given Data and Cost Calculation Formulas
- Aquarium Cost: [tex]$14.50$[/tex] per ticket
- Carnival Cost: [tex]\( C = 15.5 + 12f \)[/tex] where [tex]\( f \)[/tex] is the number of friends
- Wave Pool Cost: [tex]$16.75$[/tex] per ticket for groups of 4 or fewer, [tex]$12.25$[/tex] per ticket for groups larger than 4
The table provided is:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{\# of friends} & \text{Carnival Cost} & \text{Aquarium Cost} & \text{Wave Pool Cost} \\ \hline 3 & 51.5 & 43.5 & 50.25 \\ \hline 5 & 75.5 & 72.5 & 61.25 \\ \hline 7 & 99.5 & 101.5 & 85.75 \\ \hline 9 & 123.5 & 130.5 & 110.25 \\ \hline \end{array} \][/tex]
### Calculations
1. Aquarium Costs:
- For 3 friends: [tex]\( 14.5 \times 3 = 43.5 \)[/tex]
- For 5 friends: [tex]\( 14.5 \times 5 = 72.5 \)[/tex]
- For 7 friends: [tex]\( 14.5 \times 7 = 101.5 \)[/tex]
- For 9 friends: [tex]\( 14.5 \times 9 = 130.5 \)[/tex]
These figures match the values in the table.
2. Carnival Costs:
- For 3 friends: [tex]\( 15.5 + 12 \times 3 = 15.5 + 36 = 51.5 \)[/tex]
- For 5 friends: [tex]\( 15.5 + 12 \times 5 = 15.5 + 60 = 75.5 \)[/tex]
- For 7 friends: [tex]\( 15.5 + 12 \times 7 = 15.5 + 84 = 99.5 \)[/tex]
- For 9 friends: [tex]\( 15.5 + 12 \times 9 = 15.5 + 108 = 123.5 \)[/tex]
These figures also match the values in the table.
3. Wave Pool Costs:
- For 3 friends: [tex]\( 16.75 \times 3 = 50.25 \)[/tex] (since the group is 4 or fewer)
- For 5 friends: [tex]\( 12.25 \times 5 = 61.25 \)[/tex] (since the group is larger than 4)
- For 7 friends: [tex]\( 12.25 \times 7 = 85.75 \)[/tex]
- For 9 friends: [tex]\( 12.25 \times 9 = 110.25 \)[/tex]
These figures also match the values in the table.
### Verifying Gale's Claims
1. The table represents all possibilities.
Since all calculated costs match the given data in the table, this claim is true.
2. The carnival always costs the least.
Comparing the costs for each number of friends:
- For 3 friends: [tex]\( \text{Aquarium} = 43.5 \)[/tex], [tex]\( \text{Carnival} = 51.5 \)[/tex], [tex]\( \text{Wave Pool} = 50.25 \)[/tex]
- Least: Aquarium
- For 5 friends: [tex]\( \text{Aquarium} = 72.5 \)[/tex], [tex]\( \text{Carnival} = 75.5 \)[/tex], [tex]\( \text{Wave Pool} = 61.25 \)[/tex]
- Least: Wave Pool
- For 7 friends: [tex]\( \text{Aquarium} = 101.5 \)[/tex], [tex]\( \text{Carnival} = 99.5 \)[/tex], [tex]\( \text{Wave Pool} = 85.75 \)[/tex]
- Least: Wave Pool
- For 9 friends: [tex]\( \text{Aquarium} = 130.5 \)[/tex], [tex]\( \text{Carnival} = 123.5 \)[/tex], [tex]\( \text{Wave Pool} = 110.25 \)[/tex]
- Least: Wave Pool
Since the carnival is not always the least expensive, this claim is false.
3. The Aquarium always costs the most.
Comparing the costs for each number of friends:
- For 3 friends: [tex]\( \text{Aquarium} = 43.5 \)[/tex], [tex]\( \text{Carnival} = 51.5 \)[/tex], [tex]\( \text{Wave Pool} = 50.25 \)[/tex]
- Most: Carnival
- For 5 friends: [tex]\( \text{Aquarium} = 72.5 \)[/tex], [tex]\( \text{Carnival} = 75.5 \)[/tex], [tex]\( \text{Wave Pool} = 61.25 \)[/tex]
- Most: Carnival
- For 7 friends: [tex]\( \text{Aquarium} = 101.5 \)[/tex], [tex]\( \text{Carnival} = 99.5 \)[/tex], [tex]\( \text{Wave Pool} = 85.75 \)[/tex]
- Most: Aquarium
- For 9 friends: [tex]\( \text{Aquarium} = 130.5 \)[/tex], [tex]\( \text{Carnival} = 123.5 \)[/tex], [tex]\( \text{Wave Pool} = 110.25 \)[/tex]
- Most: Aquarium
Since the aquarium is not always the most expensive, this claim is false.
4. (7, 101.5) is an ordered pair used to represent the aquarium cost.
For 7 friends, the aquarium cost is indeed 101.5. Thus, this claim is true.
### Conclusion
The evaluations of Gale's claims are:
1. The table represents all possibilities: True
2. The carnival always costs the least: False
3. The Aquarium always costs the most: False
4. (7, 101.5) is an ordered pair used to represent the aquarium cost: True
### Given Data and Cost Calculation Formulas
- Aquarium Cost: [tex]$14.50$[/tex] per ticket
- Carnival Cost: [tex]\( C = 15.5 + 12f \)[/tex] where [tex]\( f \)[/tex] is the number of friends
- Wave Pool Cost: [tex]$16.75$[/tex] per ticket for groups of 4 or fewer, [tex]$12.25$[/tex] per ticket for groups larger than 4
The table provided is:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{\# of friends} & \text{Carnival Cost} & \text{Aquarium Cost} & \text{Wave Pool Cost} \\ \hline 3 & 51.5 & 43.5 & 50.25 \\ \hline 5 & 75.5 & 72.5 & 61.25 \\ \hline 7 & 99.5 & 101.5 & 85.75 \\ \hline 9 & 123.5 & 130.5 & 110.25 \\ \hline \end{array} \][/tex]
### Calculations
1. Aquarium Costs:
- For 3 friends: [tex]\( 14.5 \times 3 = 43.5 \)[/tex]
- For 5 friends: [tex]\( 14.5 \times 5 = 72.5 \)[/tex]
- For 7 friends: [tex]\( 14.5 \times 7 = 101.5 \)[/tex]
- For 9 friends: [tex]\( 14.5 \times 9 = 130.5 \)[/tex]
These figures match the values in the table.
2. Carnival Costs:
- For 3 friends: [tex]\( 15.5 + 12 \times 3 = 15.5 + 36 = 51.5 \)[/tex]
- For 5 friends: [tex]\( 15.5 + 12 \times 5 = 15.5 + 60 = 75.5 \)[/tex]
- For 7 friends: [tex]\( 15.5 + 12 \times 7 = 15.5 + 84 = 99.5 \)[/tex]
- For 9 friends: [tex]\( 15.5 + 12 \times 9 = 15.5 + 108 = 123.5 \)[/tex]
These figures also match the values in the table.
3. Wave Pool Costs:
- For 3 friends: [tex]\( 16.75 \times 3 = 50.25 \)[/tex] (since the group is 4 or fewer)
- For 5 friends: [tex]\( 12.25 \times 5 = 61.25 \)[/tex] (since the group is larger than 4)
- For 7 friends: [tex]\( 12.25 \times 7 = 85.75 \)[/tex]
- For 9 friends: [tex]\( 12.25 \times 9 = 110.25 \)[/tex]
These figures also match the values in the table.
### Verifying Gale's Claims
1. The table represents all possibilities.
Since all calculated costs match the given data in the table, this claim is true.
2. The carnival always costs the least.
Comparing the costs for each number of friends:
- For 3 friends: [tex]\( \text{Aquarium} = 43.5 \)[/tex], [tex]\( \text{Carnival} = 51.5 \)[/tex], [tex]\( \text{Wave Pool} = 50.25 \)[/tex]
- Least: Aquarium
- For 5 friends: [tex]\( \text{Aquarium} = 72.5 \)[/tex], [tex]\( \text{Carnival} = 75.5 \)[/tex], [tex]\( \text{Wave Pool} = 61.25 \)[/tex]
- Least: Wave Pool
- For 7 friends: [tex]\( \text{Aquarium} = 101.5 \)[/tex], [tex]\( \text{Carnival} = 99.5 \)[/tex], [tex]\( \text{Wave Pool} = 85.75 \)[/tex]
- Least: Wave Pool
- For 9 friends: [tex]\( \text{Aquarium} = 130.5 \)[/tex], [tex]\( \text{Carnival} = 123.5 \)[/tex], [tex]\( \text{Wave Pool} = 110.25 \)[/tex]
- Least: Wave Pool
Since the carnival is not always the least expensive, this claim is false.
3. The Aquarium always costs the most.
Comparing the costs for each number of friends:
- For 3 friends: [tex]\( \text{Aquarium} = 43.5 \)[/tex], [tex]\( \text{Carnival} = 51.5 \)[/tex], [tex]\( \text{Wave Pool} = 50.25 \)[/tex]
- Most: Carnival
- For 5 friends: [tex]\( \text{Aquarium} = 72.5 \)[/tex], [tex]\( \text{Carnival} = 75.5 \)[/tex], [tex]\( \text{Wave Pool} = 61.25 \)[/tex]
- Most: Carnival
- For 7 friends: [tex]\( \text{Aquarium} = 101.5 \)[/tex], [tex]\( \text{Carnival} = 99.5 \)[/tex], [tex]\( \text{Wave Pool} = 85.75 \)[/tex]
- Most: Aquarium
- For 9 friends: [tex]\( \text{Aquarium} = 130.5 \)[/tex], [tex]\( \text{Carnival} = 123.5 \)[/tex], [tex]\( \text{Wave Pool} = 110.25 \)[/tex]
- Most: Aquarium
Since the aquarium is not always the most expensive, this claim is false.
4. (7, 101.5) is an ordered pair used to represent the aquarium cost.
For 7 friends, the aquarium cost is indeed 101.5. Thus, this claim is true.
### Conclusion
The evaluations of Gale's claims are:
1. The table represents all possibilities: True
2. The carnival always costs the least: False
3. The Aquarium always costs the most: False
4. (7, 101.5) is an ordered pair used to represent the aquarium cost: True
