A gardener has 960 feet of fencing to fence in a rectangular garden. One side of the garden isbordered by a river and so it does not need any fencing.rivergardenWhat dimensions would guarantee that the garden has the greatest possible area?shorter side:ft (feet)longer side:ft (feet)greatest possible area:ft? (square-feet)

Name: A gardener has 960 feet of fencing to fence in a rectangular garden.
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: A gardener has 960 feet of fencing to fence in a rectangular garden. One side of the garden isbordered by a river and so it does not need any fencing.rivergardenWhat dimensions would guarantee that the garden has the greatest possible area?shorter s

Give that : A gardener has 960 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and so it does not need any fencing. Figure :By using the formula, Perimeter of rectangle = 2 ( Length + Width ) Area of rectangle = Length ×WidthHere, Fencing available = 960 feet Three sides need to be considered . We get, The perimeter of the garden would be , Perimeter = 2 L + W Perimeter = 960 is given. Then, 960 = 2L + W Solve for W :W = 960 - 2L Then area is, Area(A) = L ( 960 - 2L) = 960 L - 2L2 A = 960 L - 2L2 The above equation is quadratic equation of the the standard form f(x) = ax 2 + b x + C, we can find the coordinates of the vertex ( h, k) by the following formulas : h = -b2a , k = f(h). Compare the above equation with the standard form f(x) = ax 2 + b x + C , To get, a = -2 , b = 960, c = 0. Then, h = -b2a = -9602(-2) =9604 = 240 Plug h = 240 in the equation A = 960 L - 2L2 , A ( 240) = 960 ( 240) - 2 (240)2 = 230,400 - 115,200 = 115,200 To get, k = A( 240) = 115, 240. Then, Vertex = (h, k) = ( 240, 115,200 ) That is, ( L , W) = ( 240, 115,200) Thus, The dimensions are 240 feet by 115,200 feet. Dimension would guarantee that the garden has the greatest possible area = 240 feet by 115,200 feet. Greatest possible area = 27,648,000 ft2 . Therefore, Shorter side = 240 feet Longer side = 115,200 feet Greatest possible area = 27,648,000 ft2 .