# Problem :A man wants to construct a rectangular enclosure around his house. Only three sides must be fenced, since his garage wall will form.the fourth side. If he uses 24 meters of fencing,...

A man wants to construct a rectangular enclosure around his house. Only three sides must be fenced, since his garage wall will form.the fourth side. If he uses 24 meters of fencing, what is the maximum area possible ?

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We'll write the area of a rectangular shape.

A = width*length

A = y*x

Since only three sides of the rectangular shape must be fenced and he uses 24m of fencing, we'll write the perimeter of the fenced shape:

24 = 2y + x

We'll use symmetric property and we'll get:

x = 24 - 2y (1)

Now, we'll write the area with respect to y:

A(y) = x*y

A(y) = (24 - 2y)*y

A(y) = 24y - 2y^2

When the function A(y) has a maximum, the derivative of the function A(y) is cancelling.

A'(y) = (24y - 2y^2)'

A'(y) = 24 - 4y

A'(y) = 0

24 - 4y = 0

-4y = -24

y = -24/-4

y = 6 meters

We'll substitute y = 6 in (1):

x = 24 - 2y

x = 24 - 12

x = 12 meters

The area is:

A = x*y

A = 12*6

The maximum area is A = 72 square meters.