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

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

$505.96\ ft$

Answer 2

The minimum amount of fencing must be used to build his pen is $505.96\; \text {ft}$.

Step-by-step explanation:

Given information:

Farmer Alex has 32 llamas each of whom needs 10000 square feet.

Now the area of rectangle is $A=x.y$

Where, $x=$ length of grazing field

And $y=$ width of the grazing field

Now, the total required area

$32000=x.y\\\\y=32000/x$

Now, find the equation of perimeter of the rectangular pen:

The perimeter $P=2x+2y$

As, the one side of pen is along with the river

$P=x+2y$

Now, find the minimum area of fencing:

So for minimum perimeter:

$P=x+2(32000/x)$

Using a graphing tool (see attached figure)

The vertex of the graph is the point of minimum perimeter.

The vertex is point (252.98,505.96)

That means :

$x=252.98$

And , minimum perimeter is $505.96\; \text {ft}$

Now the value of $y$

$y=32000/x\\y=32000/252.98\\y=126.49$

Hence, The minimum amount of fencing must be used to build his pen is $505.96\; \text {ft}$.

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