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A farmer plans to enclose two adjacent rectangular pens against a barn wall (see figure).

The farmer has 270 feet of fencing available and wants to create the largest possible area.
Create a function for the enclosed area A in terms of x.
Find the dimensions that give the maximum area.

User AdrianBR
by
8.2k points

1 Answer

0 votes

Answer: Let's assume that the pig pens need to be fenced in the way shown in the diagram above.

Then, the perimeter is given by

4

x

+

3

y

=

160

.

4

x

=

160

3

y

x

=

40

3

4

y

The area of a rectangle is given by

A

=

L

×

W

, however here we have two rectangles put together, so the total area will be given by

A

=

2

×

L

×

W

.

A

=

2

(

40

3

4

y

)

y

A

=

80

y

3

2

y

2

Now, let's differentiate this function, with respect to y, to find any critical points on the graph.

A

'

(

y

)

=

80

3

y

Setting to 0:

0

=

80

3

y

80

=

3

y

80

3

=

y

x

=

40

3

4

×

80

3

x

=

40

20

x

=

20

Hence, the dimensions that will give the maximum area are

20

by

26

2

3

feet.

A graphical check of the initial function shows that the vertex is at

(

26

2

3

,

1066

2

3

)

, which represents one of the dimensions that will give the maximum area and the maximum area, respectively.

Hopefully this helps!

Step-by-step explanation: hope this helps

User Maulik Vora
by
7.8k points
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