Given the constraints of the scenario, select all of the options that are nonviable solutions to the inequality.(3 points)

Question

Given the constraints of the scenario, select all of the options that are nonviable solutions to the inequality.(3 points)
Responses

a=12, c=2
a=12, c=2

a=5.6, c=15
a=5.6, c=15

a=18, c=−3
a=18, c=−3

a=0, c=20

GPT-4o mini · Answer

To determine which options are nonviable solutions to the inequality, we need to infer what inequality is being discussed. Without the specific inequality given, we can proceed by analyzing these potential solutions based on common scenarios.

Here are some basic scenarios that could render a solution nonviable:

If the inequality involves a variable relationship like $ a < c $, a solution where $ a $ is greater than or equal to $ c $ would be nonviable.
If the inequality has to do with values being positive or within certain bounds, any option outside those bounds might be nonviable.

Let's analyze the provided options under possible basic inequalities:

a = 12, c = 2
- If considering $ a > c $, then this is viable since $ 12 > 2 $.
a = 5.6, c = 15
- Here, $ a < c $ (5.6 < 15) is also viable.
a = 18, c = -3
- Again, $ a > c $ (18 > -3) is viable.
a = 0, c = 20
- Here, $ a < c $ (0 < 20) is also viable.

Without knowing the specific inequality, all options provided could potentially be viable, depending on the context of the inequality being tested. However, if these were judged strictly by a common inequality of $ a < c $, then the first scenario $ a = 12, c = 2 $ and the third scenario $ a = 18, c = -3 $ would be nonviable solutions.

Thus, from a mathematical interpretation based on common inequalities:

Nonviable Solutions: If $ a < c $ is the inequality:
- Candidates: a = 12, c = 2 and a = 18, c = −3 would be nonviable.

If you have a specific inequality, please share it for exact determination. Otherwise, based on common conceptual views, we would assess the answers as described.

Matt · Answer

Kalani is planning a birthday party at the science center. It will cost $120 to reserve the party, plus $11.00 per child and $6.50 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 Kalani as to how many adults and children you think she should invite to the party. Explain your recommendation.(1 point)

GPT-4o mini · Answer

To help Kalani determine how many adults and children she can invite within her budget, we first need to establish an equation representing the total cost of the party.

The total cost can be expressed as:

\[ \text{Total Cost} = 120 + 11c + 6.5a \leq 300 \]

Where:

$ c $ is the number of children,
$ a $ is the number of adults.

The inequality can be simplified to:

\[ 11c + 6.5a \leq 300 - 120 \]

\[ 11c + 6.5a \leq 180 \]

Next, we can express this relationship and analyze it under the constraints that there is at least one adult and one child at the party:

1. Minimum counts:
We know that both $ c \geq 1 $ and $ a \geq 1 $.

2. Testing possible scenarios:
We can start testing some integer combinations of adults and children that satisfy the inequality $ 11c + 6.5a \leq 180 $.

Example Scenarios:

Scenario 1: $ a = 1 $ (1 adult) \[ 11c + 6.5(1) \leq 180 \ 11c + 6.5 \leq 180 \ 11c \leq 173.5 \ c \leq \frac{173.5}{11} \approx 15.77 \Rightarrow c \leq 15 \]

So, (1 adult, 15 children) is a possible scenario.

Total Cost: $ 120 + 11(15) + 6.5(1) = 120 + 165 + 6.5 = 291.5 $ (This is under budget.)
Scenario 2: $ a = 2 $ (2 adults) \[ 11c + 6.5(2) \leq 180 \ 11c + 13 \leq 180 \ 11c \leq 167 \ c \leq \frac{167}{11} \approx 15.18 \Rightarrow c \leq 15 \]

So, (2 adults, 15 children) is also possible.

Total Cost: $ 120 + 11(15) + 6.5(2) = 120 + 165 + 13 = 298 $ (Still under budget.)
Scenario 3: $ a = 3 $ (3 adults) \[ 11c + 6.5(3) \leq 180 \ 11c + 19.5 \leq 180 \ 11c \leq 160.5 \ c \leq \frac{160.5}{11} \approx 14.59 \Rightarrow c \leq 14 \]

So, (3 adults, 14 children) is viable.

Total Cost: $ 120 + 11(14) + 6.5(3) = 120 + 154 + 19.5 = 293.5 $ (Still under budget.)

Recommendation:

After testing a few scenarios, I would recommend:

3 adults and 14 children.

Total cost will be $ 293.5 $, which is under the budget of $ 300 $, and it satisfies the constraint of having at least one adult and one child. This configuration allows for a good number of children to enjoy the birthday party while ensuring adequate supervision by the adults.