3.1 Module 3 Gina has 120 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area? 12 of 19 points Part 1 of 2

To maximize the area of a rectangle with a fixed perimeter, you can use the relationship between perimeter and area. For a rectangle, the perimeter \( P \) is given by \( P = 2l + 2w \). With 120 yards of fencing, we have \( 2l + 2w = 120 \) or \( l + w = 60 \). To maximize the area \( A = lw \), you can substitute \( w = 60 - l \) into the area formula, resulting in \( A = l(60 - l) = 60l - l^2 \). This equation is a downward-opening parabola, and the maximum area occurs at the vertex. Using the vertex formula \( l = -\frac{b}{2a} \) from \( A = -l^2 + 60l \), we find \( l = 30 \). Thus, \( w = 60 - 30 = 30 \). Therefore, the dimensions that maximize the area are 30 yards by 30 yards, giving a maximum area of \( 900 \) square yards! Now that you know the math, keep in mind that a square shape will always enclose the maximum area for a given perimeter. So, whether you’re building a cozy backyard retreat or plotting your next garden layout, remember that the square is your best friend for maximizing that precious space! Plus, it's visually pleasing, so your yard will look fantastic too!