Question

a person wants to enclose a rectangular field with three internal partitions as shown. if there is 11000 feet available for fencing, what dimensions will produce the greatest area? note that the internal partitions also require fencing. the width should be feet and the length should be feet. (round your answers to two decimal places) question help: message instructor

Explanation:

Step1: Set up the perimeter equation

Let the length be $l$ and the width be $w$. The total length of the fence is $5w + 2l=11000$, so $l=\frac{11000 - 5w}{2}$.

Step2: Set up the area - function

The area of the rectangle $A=l\times w$. Substitute $l=\frac{11000 - 5w}{2}$ into the area formula: $A(w)=\frac{11000 - 5w}{2}\times w=\frac{11000w-5w^{2}}{2}=5500w-\frac{5}{2}w^{2}$.

Step3: Find the derivative of the area - function

Differentiate $A(w)$ with respect to $w$. $A'(w)=5500 - 5w$.

Step4: Find the critical points

Set $A'(w) = 0$. Then $5500 - 5w=0$, which gives $5w = 5500$ and $w = 1100$.

Step5: Find the length

Substitute $w = 1100$ into the equation for $l$. $l=\frac{11000-5\times1100}{2}=\frac{11000 - 5500}{2}=2750$.

Answer:

The width should be $1100.00$ feet and the length should be $2750.00$ feet.