Question

antoine wants to build a rectangular enclosure for his animals. one side of the pen will be against the barn, so he needs no fence on that side. the other three sides will be enclosed with wire fencing. if antoine has 900 feet of fencing, you can find the dimensions that maximize the area of the enclosure.
a) let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). write an function for the area a of the enclosure in terms of w. (hint first write two equations with w and l and a. solve for l in one equation and substitute for l in the other).
a(w)=

b) what width w would maximize the area?
w =
ft
c) what is the maximum area?
a =
square feet

Explanation:

Step1: Set up perimeter equation

Since one - side of the rectangle is against the barn and the total length of fencing is 900 feet, we have $2w + l=900$, so $l = 900 - 2w$.

Step2: Write area function

The area of a rectangle is $A=w\times l$. Substitute $l = 900 - 2w$ into the area formula, we get $A(w)=w(900 - 2w)=900w-2w^{2}$.

Step3: Find the width for maximum area

The function $A(w)=-2w^{2}+900w$ is a quadratic function in the form $y = ax^{2}+bx + c$ with $a=-2$, $b = 900$, $c = 0$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its x - coordinate (in our case, the value of $w$ for the maximum of $A(w)$) given by $w=-\frac{b}{2a}$. So $w=-\frac{900}{2\times(-2)} = 225$ feet.

Step4: Calculate the maximum area

Substitute $w = 225$ into the area function $A(w)=900w-2w^{2}$. Then $A(225)=900\times225-2\times(225)^{2}=202500 - 2\times50625=202500 - 101250=101250$ square feet.

Answer:

a) $A(w)=900w - 2w^{2}$
b) $w = 225$
c) $A = 101250$