Question

Farmer Alex has 32 llamas each of whom needs 10000 square feet of grazing area. He wants to enclose a rectangular pen along a straight river, where he does not need fence. What is the minimum amount of fencing he must use to build his pen?

Answer 1

So hmm notice the picture below

the side of the river, needs no fencing, since the river is there
and thus the perimeter of the pen, is just w + w + l, or 2w+l

thus   $\bf \textit{area of a rectangle}\\\\ A=lw\qquad A=10000\implies 10000=lw\implies \cfrac{10000}{w}=\boxed{l} \\\\\\ \textit{perimeter of enclosed pen}\\\\ P=2w+l\implies P(w)=2w+\boxed{\cfrac{10000}{w}}$

take the derivative of P(w), set it to zero, and check for any critical points, then run a first-derivative test for any minimum

Answer 2

Step 1

we know that

the area of a rectangle is equal to

$A=xy$

where

x is the length side of the rectangle

w is the width side of the rectangle

Find the total required area

$32*10,000=32,000\ ft^{2}$

so

$32,000=xy$

$y=32,000/x$ ------> equation A

Step 2

Find the equation of the perimeter of the rectangular pen

we know that

the perimeter of the rectangle is equal to

$P=2x+2y$

Remember that  one side of the pen is along a straight river

so the perimeter is equal to

$P=x+2y$------> equation B  

Step 3

Find the minimum amount of fencing

we know that

the minimum amount of fencing is when the perimeter is a minimum

Substitute equation A in equation B

$P=x+2*(32,000/x)$      

Using a graphing tool

see the attached figure

The vertex of the graph is the point for a minimum perimeter

the vertex is the point  $(252.98,505.96)$

that means

for $x=252.98\ ft$

The minimum perimeter is equal to $505.96\ ft$

Find the value of y

$y=32,000/x$

$y=32,000/252.98$

$y=126.49\ ft$

therefore

The answer is

the minimum amount of fencing he must use to build his pen is

$505.96\ ft$