Julia plans to place 52 ft of decorative fencing around three sides of a new rectangular rose garden adjacent to her home. What dimensions will produce the maximum area for her roses, and how many square feet of garden will she have to write a quadratic function, modeling each situation and use the function to complete the exercise

1. Write the equation for the perimeter of the three sides: 2y + x = 52 .
2. Solve for \( x \): x = 52 - 2y .
3. Write the area of the garden: A = x \cdot y .
4. Substitute \( x \) in the area formula: A = (52 - 2y) \cdot y , resulting in A = 52y - 2y^2 .
5. Identify the function's coefficients: \( a = -2 \), \( b = 52 \).
6. Find \( y \) that maximizes the area using the vertex formula: y = -\frac{b}{2a} = \frac{52}{4} = 13 .
7. Find \( x \) when \( y = 13 \): x = 52 - 2 \times 13 = 26 .
8. Calculate maximum area: A = 26 \cdot 13 = 338 \text{ square feet} .
9. The maximum area is 338 square feet, with dimensions 26 ft by 13 ft.