Question

a gardener has 800 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)

Explanation:

Step1: Define variables

Let the length of the side perpendicular to the river be \(x\) (shorter side), and the length of the side parallel to the river be \(y\) (longer side). The perimeter equation (since one side is not fenced) is \(2x + y=800\), so \(y = 800 - 2x\). The area \(A=xy=x(800 - 2x)=800x-2x^{2}\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(A(x)=ax^{2}+bx + c\) (here \(a=-2\), \(b = 800\), \(c = 0\)), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). Substituting \(a=-2\) and \(b = 800\) into the formula \(x=-\frac{800}{2\times(-2)}=200\).

Step3: Find the value of \(y\)

Substitute \(x = 200\) into \(y=800 - 2x\). Then \(y=800-2\times200 = 400\).

Step4: Calculate the area

Substitute \(x = 200\) and \(y = 400\) into \(A=xy\). So \(A=200\times400=80000\).

Answer:

shorter side: \(200\) ft
longer side: \(400\) ft
greatest possible area: \(80000\) \(ft^{2}\)