Question

1. diana has 2400 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area? reference hw 3.1, #1

Explanation:

Step1: Let the length be $x$ and width be $y$.

The perimeter of the rectangle is $2(x + y)=2400$, so $x + y=1200$, and $y = 1200 - x$.

Step2: Express the area formula.

The area $A=xy=x(1200 - x)=1200x - x^{2}$.

Step3: Find the maximum of the quadratic - function.

For a quadratic function $A(x)=-x^{2}+1200x$, where $a=-1$, $b = 1200$, the vertex of the parabola $x=-\frac{b}{2a}$.
$x=-\frac{1200}{2\times(-1)} = 600$.

Step4: Find the value of $y$.

Since $y=1200 - x$, when $x = 600$, $y=1200 - 600=600$.

Step5: Calculate the maximum area.

$A = xy=600\times600 = 360000$ square yards.

Answer:

The dimensions of the rectangle are 600 yards by 600 yards, and the maximum area is 360000 square yards.