Question

anthony 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 anthony has 950 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
question help: video message instructor

Explanation:

Step1: Set up the perimeter - related equation

Since one side is against the barn and the total length of fencing is 950 feet, we have $2w + l=950$, so $l = 950 - 2w$. The area of a rectangle is $A=l\times w$.

Step2: Write the area function in terms of $w$

Substitute $l = 950 - 2w$ into the area formula $A=lw$. We get $A(w)=(950 - 2w)w=950w-2w^{2}$.

Step3: Find the maximum of the area function

The function $A(w)=- 2w^{2}+950w$ is a quadratic function of the form $y = ax^{2}+bx + c$ with $a=-2$, $b = 950$, $c = 0$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its $x$ - coordinate (in our case $w$ - coordinate) at $w=-\frac{b}{2a}$.
$w=-\frac{950}{2\times(-2)}=\frac{950}{4}=237.5$ feet.

Step4: Calculate the maximum area

Substitute $w = 237.5$ into the area function $A(w)=950w-2w^{2}$.
$A(237.5)=950\times237.5-2\times(237.5)^{2}$
$A(237.5)=226375 - 2\times56406.25$
$A(237.5)=226375-112812.5 = 113562.5$ square - feet.

Answer:

a) $A(w)=950w - 2w^{2}$
b) $w = 237.5$
c) $A = 113562.5$