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Andrea works at a zoo that just purchased a new lion and two parakeets. Andrea needs to change a section of the lion's habitat from sand to grass. She also needs to fill up the parakeets' food dispenser. Her boss told her that the length of the grassy area for the lion must be 3 feet greater than its width and that the food dispenser for the parakeets must be filled up completely. To give the lion enough grass, the area of the grass must be at least 8 times greater than the radius of the parakeets' food dispenser.

The food dispenser for the parakeets is cylindrical in shape and 4 feet tall. It is exactly halfway full. The cost of the parakeets' food is \[tex]$0.49 times the length of the radius for every foot of height of food missing from the dispenser. Also, the landscaping crew said they could put the grass in the lion's habitat for \$[/tex]2.84 per square foot of grass. Andrea cannot spend more than \$751.00 on both projects.

If [tex]\(x\)[/tex] represents the width of the grassy area and [tex]\(y\)[/tex] represents the radius of the parakeets' food dispenser, then which of the following systems of inequalities can be used to determine the length and width of the grassy area and the radius of the food dispenser?

A. [tex]\(\left\{\begin{array}{l}y \leq 0.13 x^2 + 0.38 x \\ y \leq 766.33 - 2.90 x^2 - 8.69 x\end{array}\right.\)[/tex]

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]

C. [tex]\(\left\{\begin{array}{l}y \leq 8 x^2 + 24 x \\ y \leq 383.16 - 1.45 x^2 - 4.35 x\end{array}\right.\)[/tex]



Answer :

GinnyAnswer
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.