diferira7c

Answered

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?

Answer & Explanation

amarantha41

Expert

2021-12-26Added 38 answers

To get the dimensions of our rectangle, we must first determine x and y.

We can see from the sketch that we have

$2x+3y=400\Rightarrow x=200-\frac{3}{2}y$

Our area is

$A(x,y)=2xy$

As we can see from the first equation, our function is

$A(x,y)=2xy$

We can see from the first equation that we have So our function is

$A(x,y)=2xy$

$A\left(y\right)=2\cdot (200-\frac{3}{2}y)\cdot y$

$A\left(y\right)=(400-3y)y$

$\Rightarrow A\left(y\right)=-3{y}^{2}+400y$

The first step is to identify critical numbers and solve them -

${A}^{\prime}\left(y\right)=0$

This leads to

${A}^{\prime}\left(y\right)=-6y+400$

So our equation becomes

$-6y+400=0\Rightarrow y=\frac{200}{3}$

When we plug this into expression for x, we get

$x=200-\frac{3}{2}y\Rightarrow x=100$

As a result, our ultimate dimensions are

$x=100;y=\frac{200}{3}$

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