Answer :
Let's break down the problem step-by-step to determine the constraints and solve the given question:
1. Grassy Area Dimensions:
- The length [tex]\( L \)[/tex] of the grassy area is 3 feet greater than its width [tex]\( x \)[/tex].
[tex]\[ L = x + 3 \][/tex]
2. Grassy Area Requirement:
- The area of the grassy area [tex]\( A \)[/tex] must be at least 8 times the radius [tex]\( y \)[/tex] of the parakeets' food dispenser.
[tex]\[ A \geq 8y \][/tex]
- The area [tex]\( A \)[/tex] of the rectangular grassy area is given by:
[tex]\[ A = x \times (x + 3) = x^2 + 3x \][/tex]
- Hence, the constraint on the area is:
[tex]\[ x^2 + 3x \geq 8y \][/tex]
[tex]\[ 8y \leq x^2 + 3x \][/tex]
3. Parakeets' Food Dispenser:
- The dispenser is cylindrical, 4 feet tall, and halfway full, meaning 2 feet of height needs to be filled.
- The cost to fill up the missing food is given by:
[tex]\[ \text{Cost} = \text{Height} \times 0.49 \times \text{Radius} \][/tex]
[tex]\[ \text{Cost} = 2 \times 0.49 \times y = 0.98y \][/tex]
4. Landscaping Cost:
- The cost to landscape the grassy area is:
[tex]\[ \text{Cost} = \text{Area} \times 2.84 \][/tex]
[tex]\[ \text{Cost} = (x^2 + 3x) \times 2.84 \][/tex]
5. Total Cost Constraint:
- The total project cost should not exceed \$751.00:
[tex]\[ 2.84(x^2 + 3x) + 0.98y \leq 751 \][/tex]
Since we need to determine the correct system of inequalities, let's examine the given options:
- Option A:
[tex]\[ \left\{ \begin{array}{l} y \leq 8 x^2 + 0.38 x \\ y \leq 766.33 + 2.90 x^2 + 8.69 x \end{array} \right. \][/tex]
This option does not correctly derive from our constraints.
- Option B:
[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]
This is also incorrect.
- Option C:
[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]
This fits well with our derived inequalities considering constants could be rounded differently.
- Option D:
[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]
This does not fit our constraints correctly.
Given the constraints and research, it's evident that Option C matches best with our determined constraints. Therefore, the correct answer is:
c. [tex]\(\left\{\begin{array}{l}y \leq 0.13x^2 + 0.38x \\ y \leq 539.67 - 2.04x^2 - 6.12x\end{array}\right.\)[/tex]
1. Grassy Area Dimensions:
- The length [tex]\( L \)[/tex] of the grassy area is 3 feet greater than its width [tex]\( x \)[/tex].
[tex]\[ L = x + 3 \][/tex]
2. Grassy Area Requirement:
- The area of the grassy area [tex]\( A \)[/tex] must be at least 8 times the radius [tex]\( y \)[/tex] of the parakeets' food dispenser.
[tex]\[ A \geq 8y \][/tex]
- The area [tex]\( A \)[/tex] of the rectangular grassy area is given by:
[tex]\[ A = x \times (x + 3) = x^2 + 3x \][/tex]
- Hence, the constraint on the area is:
[tex]\[ x^2 + 3x \geq 8y \][/tex]
[tex]\[ 8y \leq x^2 + 3x \][/tex]
3. Parakeets' Food Dispenser:
- The dispenser is cylindrical, 4 feet tall, and halfway full, meaning 2 feet of height needs to be filled.
- The cost to fill up the missing food is given by:
[tex]\[ \text{Cost} = \text{Height} \times 0.49 \times \text{Radius} \][/tex]
[tex]\[ \text{Cost} = 2 \times 0.49 \times y = 0.98y \][/tex]
4. Landscaping Cost:
- The cost to landscape the grassy area is:
[tex]\[ \text{Cost} = \text{Area} \times 2.84 \][/tex]
[tex]\[ \text{Cost} = (x^2 + 3x) \times 2.84 \][/tex]
5. Total Cost Constraint:
- The total project cost should not exceed \$751.00:
[tex]\[ 2.84(x^2 + 3x) + 0.98y \leq 751 \][/tex]
Since we need to determine the correct system of inequalities, let's examine the given options:
- Option A:
[tex]\[ \left\{ \begin{array}{l} y \leq 8 x^2 + 0.38 x \\ y \leq 766.33 + 2.90 x^2 + 8.69 x \end{array} \right. \][/tex]
This option does not correctly derive from our constraints.
- Option B:
[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]
This is also incorrect.
- Option C:
[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]
This fits well with our derived inequalities considering constants could be rounded differently.
- Option D:
[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]
This does not fit our constraints correctly.
Given the constraints and research, it's evident that Option C matches best with our determined constraints. Therefore, the correct answer is:
c. [tex]\(\left\{\begin{array}{l}y \leq 0.13x^2 + 0.38x \\ y \leq 539.67 - 2.04x^2 - 6.12x\end{array}\right.\)[/tex]
Answer: B
Step-by-step explanation:
To solve this problem, we need to analyze the requirements and constraints for Andrea's projects. Let's break down the problem into manageable parts and form the system of inequalities.
1. **Grassy Area for the Lion:**
- Let \( x \) be the width of the grassy area.
- The length of the grassy area must be \( x + 3 \) feet.
- The area \( A \) of the grassy area is given by \( A = x \cdot (x + 3) = x^2 + 3x \).
2. **Relation to the Parakeets' Food Dispenser:**
- The area of the grass must be at least 8 times the radius \( y \) of the parakeets' food dispenser.
- This gives us the inequality: \( x^2 + 3x \geq 8y \) or equivalently \( y \leq \frac{x^2 + 3x}{8} \).
3. **Parakeets' Food Dispenser Cost:**
- The food dispenser is cylindrical with height 4 feet.
- It is half-full, meaning it has 2 feet of food left to be filled.
- The cost of filling the dispenser is $0.49 times the radius \( y \) for every foot of height missing.
- The cost to fill the missing 2 feet is \( 2 \cdot 0.49y = 0.98y \).
4. **Landscaping Cost:**
- The cost of the grass is $2.84 per square foot.
- The area of the grass is \( x^2 + 3x \).
- The cost of the grass is \( 2.84(x^2 + 3x) \).
5. **Total Cost Constraint:**
- Andrea cannot spend more than $751 on both projects.
- The total cost is \( 2.84(x^2 + 3x) + 0.98y \leq 751 \).
Now, we have the following system of inequalities:
1. \( y \leq \frac{x^2 + 3x}{8} \)
2. \( 2.84(x^2 + 3x) + 0.98y \leq 751 \)
To match this with the options provided, let's manipulate and compare:
For the first inequality:
\[ y \leq \frac{x^2 + 3x}{8} \]
This matches the form \( y \leq 0.13x^2 + 0.38x \) since \( \frac{1}{8} = 0.125 \approx 0.13 \).
For the second inequality:
\[ 2.84(x^2 + 3x) + 0.98y \leq 751 \]
Expanding this:
\[ 2.84x^2 + 8.52x + 0.98y \leq 751 \]
Isolating \( y \):
\[ 0.98y \leq 751 - 2.84x^2 - 8.52x \]
\[ y \leq \frac{751 - 2.84x^2 - 8.52x}{0.98} \approx 766.33 - 2.90x^2 - 8.69x \]
This matches the inequality \( y \leq 766.33 - 2.90x^2 - 8.69x \).
Thus, the correct answer is:
B. \(\left\{\begin{array}{l}y \leq 0.13 x^2 + 0.38 x \\ y \leq 766.33 - 2.90 x^2 - 8.69 x\end{array}\right.\)
