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? Enter only the maximum area. Do not include units in your answer.

Answer 1

The dimensions of is width: 41, length: 82

Explanation:

Consider the provided information.

Let y is the length of side parallel to the wall and x is length of each side perpendicular to the wall.

The perimeter of rectangle is 2x + 2y since, we need to use the side of building and fencing for the other 3 sides. Therefore,

y + 2x = 164

y = 164-2x

The area of rectangle is xy.

`A = xy = x(164-2x)`

`A = -2x^2+164x`

The above equation is in the form of a quadratic equation
`ax^2+bx+c=0`.

The graph of the function is a parabola opening downward. As the coefficient of x² is negative. The maximum occurs at the x-coordinate of the vertex.

In order to find the vertex, use the formula
`x=(-b)/(2a)[tex] and substitute the value of x in above equation.</p><p>x = -164/(2(-2)) = 41</p><p>Now substitute x = 41 in [tex]A = -2x^2+164x`

`A = -2(41)^2+164(41)`

`A = -2(1681)+6724`

`A = -3362+6724`

`A = 3362`

So the vertex is (41,3362).

This shows us that the max area is then 3362 square feet.

Now substitute the value of x in y = 164-2x

y = 164-2(41)=82

Hence, the dimensions of is width: 41, length: 82