You have 42 ft. of fencing (1 ft. segments) to make a rectangular garden. How much should each side be to maximize your total area? What will the total area be for the garden? Find the length, width, and area of your garden that will maximize its total area. Length:Width: Area:

Answer:[tex]Length=10.5\ ft[/tex][tex]Width=10.5\ ft[/tex][tex]Area=110.25\ ft^{2}[/tex]Step-by-step explanation:Letx----> the length of the rectangular gardeny---> the width of the rectangular gardenwe know thatThe perimeter of the rectangle is equal to[tex]P=2(x+y)[/tex]we have[tex]P=42\ ft[/tex]so[tex]42=2(x+y)[/tex]simplify[tex]21=(x+y)[/tex][tex]y=21-x[/tex]------> equation ARemember that the area of rectangle is equal to[tex]A=xy[/tex] ----> equation Bsubstitute equation A in equation B[tex]A=x(21-x)[/tex] [tex]A=21x-x^{2}[/tex]----> this is a vertical parabola open downwardThe vertex is a maximumThe y-coordinate of the vertex is the maximum areaThe x-coordinate of the vertex is the length side of the rectangle that maximize the areausing a graphing toolThe vertex is the point [tex](10.5,110.25)[/tex]see the attached figureso[tex]x=10.5\ ft[/tex]Find the value of y[tex]y=21-10.5=10.5\ ft[/tex]The garden is a squarethe area is equal to[tex]A=(10.5)(10.5)=110.25\ ft^{2}[/tex] ----> is equal to the y-coordinate of the vertex is correct