To determine which inequalities and bounds apply to this situation, let's break down the problem step by step:
1. Understanding the Problem:
- Mia has 50 feet of fencing and uses part of her house as one side of the kennel.
- Therefore, only three sides of the kennel need to be fenced.
- We need to find the possible length ([tex]\( l \)[/tex]) of the kennel such that the enclosed area is at least 300 square feet.
2. Setting up the Equations:
- Let's denote the length of the kennel as [tex]\( l \)[/tex].
- The width of the kennel will then be [tex]\( 50 - 2l \)[/tex], because the fencing has to cover two lengths plus one width and we have a total of 50 feet available for fencing.
- However, since this is a specific question, it asks for certain inequalities and bounds we can setup different equations considering the options provided.
3. Inequality for Enclosed Area:
- The area [tex]\( A \)[/tex] of the kennel is given by multiplying the length and width:
[tex]\[ A = l \times (50 - 2l) \][/tex]
- Since the area has to be at least 300 square feet, we set up the inequality:
[tex]\[ l \times (50 - 2l) \geq 300 \][/tex]
4. Other Provided Inequalities:
- There is another possible format given in the options:
[tex]\[ (50 + 2l) \times l < 300 \][/tex]
5. Checking Length Bounds:
- We are given several bounds for [tex]\( l \)[/tex]:
[tex]\[ 10 \leq l \leq 15 \][/tex]
[tex]\[ 0 \leq l \leq 30 \][/tex]
[tex]\[ 0 \leq l \leq 5 \][/tex]
6. Identifying Correct Inequalities and Solutions:
- The proper inequality representing the enclosed area requirement is:
[tex]\[ l(50 - 2l) \geq 300 \][/tex]
- The other inequality is:
[tex]\[ (50 + 2l)l < 300 \][/tex]
- Among the provided bounds, we need to determine which are correct based on the solution set:
- For the equation [tex]\( l(50 - 2l) \geq 300 \)[/tex], the valid solutions for [tex]\( l \)[/tex] are:
[tex]\[ 10 \leq l \leq 15 \][/tex]
- For general bounds of [tex]\( l \)[/tex]:
[tex]\[ 0 \leq l \leq 30 \][/tex]
[tex]\[ 0 \leq l \leq 5 \][/tex]
- The condition [tex]\( 0 \leq l \leq 5 \)[/tex] also fits within the solution set defined by the earlier calculations.
7. Final Selection of Options:
- The correct inequalities and bounds that represent this situation are:
- [tex]\( l(50 - 2l) \geq 300 \)[/tex]
- [tex]\( (50 + 2l)l < 300 \)[/tex]
- [tex]\( 10 \leq l \leq 15 \)[/tex]
- [tex]\( 0 \leq l \leq 30 \)[/tex]
- [tex]\( 0 \leq l \leq 5 \)[/tex]
These inequalities and bounds ensure that the kennel has at least 300 square feet of enclosed area and adhere to the constraints provided by the available fencing.
