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 maximize the area of Charlie's rectangular rose garden, he needs to express the area \( A \) in terms of one of the sides, say the length \( l \). Given that the perimeter is 20 m, the relationship between the length, width \( w \), and the perimeter is \( 2l + 2w = 20 \), which simplifies to \( w = 10 - l \). Thus, the area can be expressed as \( A(l) = l(10 - l) = 10l - l^2 \), a quadratic function where the area depends on the length of one side. To find the maximum area, Charlie should look for the vertex of this quadratic function. The vertex form provides the maximum point at \( l = \frac{-b}{2a} \). Here, \( a = -1 \) and \( b = 10 \), leading to \( l = \frac{10}{2} = 5 \). Plugging this length back into the equation for width gives him a perfect square garden of dimensions \( 5 m \) x \( 5 m \), yielding a maximum area of \( 25 m^2 \). Voilà!