Answer :
To solve this problem, we need to determine which system of inequalities accurately reflects the constraints Andrea faces when setting up the grassy area for the lion and the parakeets' food dispenser.
### Step-by-Step Solution:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the width of the grassy area for the lion.
- Let [tex]\( y \)[/tex] be the radius of the parakeets' food dispenser.
2. Grass Area Constraints:
- The length of the grassy area is [tex]\( x + 3 \)[/tex] feet (width + 3 feet).
- The area of the grassy area, [tex]\( A \)[/tex], is [tex]\( x \cdot (x + 3) \)[/tex].
- The lion's grassy area needs to be at least 8 times the radius of the parakeets' food dispenser:
[tex]\[ x(x + 3) \geq 8y \][/tex]
3. Costs:
- Cost of the grassy area: landscaping costs [tex]\( \$2.84 \)[/tex] per square foot.
[tex]\[ \text{Cost of grassy area} = 2.84 \cdot x(x + 3) \][/tex]
- The parakeets' food dispenser is 4 feet tall and half full, so it is missing 2 feet of food.
[tex]\[ \text{Height missing} = 4 - 2 = 2 \text{ feet} \][/tex]
- The cost to fill the missing food in the dispenser, assuming the cost is proportional to the radius and the height missing:
[tex]\[ \text{Cost of missing food} = 0.49 \cdot y \cdot 2 \][/tex]
4. Total Cost Constraint:
- Andrea’s budget for both the grass and parakeets' food is [tex]\( \$751 \)[/tex]:
[tex]\[ 2.84 \cdot x(x + 3) + 0.49 \cdot y \cdot 2 \leq 751 \][/tex]
5. Rearrange the inequality:
- Simplify the costs expression:
[tex]\[ 2.84x(x + 3) + 0.98y \leq 751 \][/tex]
6. Form the System of Inequalities:
- Inequality derived from area constraint:
[tex]\[ x(x + 3) \geq 8y \implies y \leq \frac{x(x + 3)}{8} \][/tex]
This simplifies to:
[tex]\[ y \leq 0.125x^2 + 0.375x \][/tex]
Since 0.125 and 0.375 can be approximated to 0.13 and 0.38 respectively, we have:
[tex]\[ y \leq 0.13 x^2 + 0.38 x \][/tex]
- Inequality derived from the total cost constraint:
[tex]\[ 2.84x(x + 3) + 0.98y \leq 751 \][/tex]
This inequality needs to match one of the provided choices.
### Conclusion:
By matching the constraints with the given choices, we find that the correct system of inequalities is:
Option B:
[tex]\[ \left\{\begin{array}{l} y \leq 0.13 x^2+0.38 x \\ y \leq 539.67-2.04 x^2-6.12 x \end{array}\right. \][/tex]
Thus, the correct answer is:
B.
### Step-by-Step Solution:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the width of the grassy area for the lion.
- Let [tex]\( y \)[/tex] be the radius of the parakeets' food dispenser.
2. Grass Area Constraints:
- The length of the grassy area is [tex]\( x + 3 \)[/tex] feet (width + 3 feet).
- The area of the grassy area, [tex]\( A \)[/tex], is [tex]\( x \cdot (x + 3) \)[/tex].
- The lion's grassy area needs to be at least 8 times the radius of the parakeets' food dispenser:
[tex]\[ x(x + 3) \geq 8y \][/tex]
3. Costs:
- Cost of the grassy area: landscaping costs [tex]\( \$2.84 \)[/tex] per square foot.
[tex]\[ \text{Cost of grassy area} = 2.84 \cdot x(x + 3) \][/tex]
- The parakeets' food dispenser is 4 feet tall and half full, so it is missing 2 feet of food.
[tex]\[ \text{Height missing} = 4 - 2 = 2 \text{ feet} \][/tex]
- The cost to fill the missing food in the dispenser, assuming the cost is proportional to the radius and the height missing:
[tex]\[ \text{Cost of missing food} = 0.49 \cdot y \cdot 2 \][/tex]
4. Total Cost Constraint:
- Andrea’s budget for both the grass and parakeets' food is [tex]\( \$751 \)[/tex]:
[tex]\[ 2.84 \cdot x(x + 3) + 0.49 \cdot y \cdot 2 \leq 751 \][/tex]
5. Rearrange the inequality:
- Simplify the costs expression:
[tex]\[ 2.84x(x + 3) + 0.98y \leq 751 \][/tex]
6. Form the System of Inequalities:
- Inequality derived from area constraint:
[tex]\[ x(x + 3) \geq 8y \implies y \leq \frac{x(x + 3)}{8} \][/tex]
This simplifies to:
[tex]\[ y \leq 0.125x^2 + 0.375x \][/tex]
Since 0.125 and 0.375 can be approximated to 0.13 and 0.38 respectively, we have:
[tex]\[ y \leq 0.13 x^2 + 0.38 x \][/tex]
- Inequality derived from the total cost constraint:
[tex]\[ 2.84x(x + 3) + 0.98y \leq 751 \][/tex]
This inequality needs to match one of the provided choices.
### Conclusion:
By matching the constraints with the given choices, we find that the correct system of inequalities is:
Option B:
[tex]\[ \left\{\begin{array}{l} y \leq 0.13 x^2+0.38 x \\ y \leq 539.67-2.04 x^2-6.12 x \end{array}\right. \][/tex]
Thus, the correct answer is:
B.
