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14 days ago

9

Farmer Alex has 32 llamas each of whom needs 10000 square feet of grazing area. He wants to enclose a rectangular pen along a st

raight river, where he does not need fence. What is the minimum amount of fencing he must use to build his pen?

2 answers:

tester [8.8K]14 days ago

8 0

Take note of the image below

the riverbank requires no fencing due to the river's presence
so the pen's perimeter can be calculated as 2w + l, or w + w + l
thus $\bf \textit{area of a rectangle}\\\\
A=lw\qquad A=10000\implies 10000=lw\implies \cfrac{10000}{w}=\boxed{l}
\\\\\\
\textit{perimeter of enclosed pen}\\\\
P=2w+l\implies P(w)=2w+\boxed{\cfrac{10000}{w}}$

derive P(w), set it to zero, locate any critical points, and perform a first-derivative test for minimum values.

Inessa [9K]14 days ago

6 0

Step 1

it is established that

the area of a rectangle can be expressed as

$A=xy$

where

x signifies the length and

w indicates the width of the rectangle

Determine the total necessary area

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

therefore

---> equation A

$32,000=xy$

Step 2

$y=32,000/x$

Derive the equation for the perimeter of the rectangular pen

it is known that

the perimeter of the rectangle is equal to

Remember that one side of the pen is adjacent to the river

thus, the perimeter is given by

$P=2x+2y$ ---> equation B

Step 3

Determine the minimum fencing requirement

it is acknowledged that $P=x+2y$

the lowest amount of fencing occurs when the perimeter is minimized

Substituting equation A into equation B

Utilizing a graphing tool

refer to the attached image

The vertex of the graph represents the point for minimized perimeter

the vertex is located at

$P=x+2*(32,000/x)$

this signifies

for

The minimum perimeter equals

$(252.98,505.96)$ Ascertain the value of y

$x=252.98\ ft$

$505.96\ ft$

$y=32,000/x$

thus

$y=32,000/252.98$

The solution is

$y=126.49\ ft$

the least amount of fencing required to construct his pen is

$505.96\ ft$

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