Farmer alex has 32 llamas each of whom needs 1 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?

The minimum amount of fencing must be used to build his pen is .

Step-by-step explanation:

Given information:

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

Now the area of rectangle is

Where, length of grazing field

And width of the grazing field

Now, the total required area

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

The perimeter

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

Now, find the minimum area of fencing:

So for minimum perimeter:

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 :

And , minimum perimeter is

Now the value of

Hence, The minimum amount of fencing must be used to build his pen is .

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Step 1

we know that

the area of a rectangle is equal to

where

x is the length side of the rectangle

w is the width side of the rectangle

Find the total required area

so

------> 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

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

so the perimeter is equal to

------> 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

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  

that means

for

The minimum perimeter is equal to

Find the value of y

therefore

The answer is

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