Question

A rectangular patio is to be constructed using the side of a building as one side and fencing for the other three sides. There are 164 feet of fencing available. Determine the dimensions that would create the patio of maximum area. What is the maximum area

Answer 1

Answer:

Therefore the dimension of the patio is 82 feet by 41 feet.

The maximum area of the rectangular patio is 3,362 square feet.

Step-by-step explanation:

Given that, a building is constructed by using one side of the rectangular patio and facing for other 3 sides.

The length of facing is = 164 feet.

Let the length of the the rectangular patio be x which is also one side of the building.

And the breadth of the rectangular patio be y.

The length of facing

=Perimeter of the patio - common side of the building and the patio

=2(x+y)-x

=2x+2y-x

=2y+x

According to the problem,

2y+x=164

$\Rightarrow x=164-2y$

The area of the rectangular patio

A= xy

Putting the value of x

A= (164-2y)y

$\Rightarrow A=164y -2y^2$

Differentiating with respect to y

A'= 164-4y

Again differentiating with respect to y

A''= - 4

To find the maximum area we set A'=0

∴164-4y=0

⇒4y=164

⇒y=41.

Now   $A''|_{y=41}=-4<0$

Since at y=41, A''<0, Then the area of the patio will be maximum when y=41.

Then ,

x=164-(2.41)

 =82 feet

Therefore the dimension of the patio is 82 feet by 41 feet.

The area of the patio = (82×41) square feet

                                  = 3362 square feet

The maximum area of the rectangular patio is 3,362 square feet.