Question

6 you have 350 yards of fencing to enclose a rectangular region. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area?

Explanation:

Step1: Define variables and perimeter

Let length = $l$, width = $w$. Perimeter: $2l + 2w = 350$ → $l + w = 175$ → $l = 175 - w$

Step2: Set up area function

Area $A(w) = w(175 - w) = -w^2 + 175w$

Step3: Find vertex of quadratic

For $ax^2+bx+c$, vertex at $w = -\frac{b}{2a} = -\frac{175}{2(-1)} = 87.5$

Step4: Calculate length and area

$l = 175 - 87.5 = 87.5$. Max area: $A = 87.5 \times 87.5$

Answer:

The dimensions of the rectangle are 87.5 yards by 87.5 yards, and the maximum area is 7656.25 square yards.