U3.L4.HW: Quadratic Functions: Question 23

Question

Explain your answers after you answer these three measurements.

A gardener has 1080 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. What dimensions would guarantee that the garden has the greatest possible area?

shorter side: _______ ft (feet)

longer side: _______ ft (feet)

greatest possible area: _________ ft²  (square-feet)

Answer 1

Determine perimeter function and solve for L

L + 2W = 1080 (remember that we exclude one side as stated in the problem!)

L = 1080 - 2W

Find the area function

A = WL

A = W(1080 - 2W)

A = 1080W - 2W²

Determine maximum width (shorter side)

W = -b/2a = -1080/(2(-2)) = -1080/(-4) = 270 feet

Determine maximum length (longer side)

L = 1080 - 2W = 1080 - 2(270) = 1080 - 540 = 540 feet

Determine maximum area given max width and length

A = WL = (270)(540) = 145,800 square-feet

Hope these answers and explanations helped!

Answer 2

Area is A = xy; 2x + y = 1080; y = 1080 - 2x, so A = x(1080 - 2x) = - 2x² + 1080x = - 2(x² - 540x + 270²) +

2 ·270² = 145800 - 2(x - 270)². From here max A = 145800 when x = 270 ft, y = 540 ft