diferira7c

2021-12-25

A rancher has 400 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?

amarantha41

Expert

To get the dimensions of our rectangle, we must first determine x and y.
We can see from the sketch that we have
$2x+3y=400⇒x=200-\frac{3}{2}y$
Our area is
$A\left(x,y\right)=2xy$
As we can see from the first equation, our function is
$A\left(x,y\right)=2xy$
We can see from the first equation that we have So our function is
$A\left(x,y\right)=2xy$
$A\left(y\right)=2\cdot \left(200-\frac{3}{2}y\right)\cdot y$
$A\left(y\right)=\left(400-3y\right)y$
$⇒A\left(y\right)=-3{y}^{2}+400y$
The first step is to identify critical numbers and solve them -
${A}^{\prime }\left(y\right)=0$
${A}^{\prime }\left(y\right)=-6y+400$
So our equation becomes
$-6y+400=0⇒y=\frac{200}{3}$
When we plug this into expression for x, we get
$x=200-\frac{3}{2}y⇒x=100$
As a result, our ultimate dimensions are
$x=100;y=\frac{200}{3}$

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