Question 8. Charlie wishes to fence off a rectangular rose garden in the middle of his backyard using 20 m of fencing material. By expressing the area as a quadratic function of the length of one side of the

To solve the problem, we need to express the area of the rectangular rose garden as a quadratic function of the length of one side. Let's denote the length of the garden as \( l \) and the width as \( w \). ### Step 1: Set up the perimeter equation The total amount of fencing material available is 20 m, which corresponds to the perimeter of the rectangle. The formula for the perimeter \( P \) of a rectangle is given by: \[ P = 2l + 2w \] Setting this equal to 20 m, we have: \[ 2l + 2w = 20 \] ### Step 2: Simplify the perimeter equation We can simplify this equation by dividing everything by 2: \[ l + w = 10 \] ### Step 3: Express width in terms of length From the equation \( l + w = 10 \), we can express \( w \) in terms of \( l \): \[ w = 10 - l \] ### Step 4: Set up the area function The area \( A \) of the rectangle is given by: \[ A = l \times w \] Substituting \( w \) from the previous step, we get: \[ A = l \times (10 - l) \] ### Step 5: Expand the area function Now, we can expand this expression: \[ A = 10l - l^2 \] ### Step 6: Write the area as a quadratic function The area \( A \) can be expressed as a quadratic function of \( l \): \[ A(l) = -l^2 + 10l \] This is a quadratic function in standard form \( A(l) = -l^2 + 10l \), where the coefficient of \( l^2 \) is negative, indicating that the parabola opens downwards. ### Conclusion The area of the rectangular rose garden as a quadratic function of the length of one side \( l \) is: \[ A(l) = -l^2 + 10l \]