Middle School

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------------------------------------------------ Tyler has 120 linear feet of fencing to put around his garden. If he uses the fence to border a rectangular garden, what is the maximum area his garden can be?

Your answer should include units such as square feet.

Answer :

Answer:

The maximum area is [tex]900\ ft^{2} [/tex]

Step-by-step explanation:

Let

x----> the length of rectangle

y---> the width of rectangle

we know that

The perimeter of rectangle is equal to

[tex]P=2(x+y)[/tex]

we have

[tex]P=120\ ft[/tex]

so

[tex]120=2(x+y)[/tex]

[tex]60=(x+y)[/tex]

[tex]y=60-x[/tex]------> equation A

Remember that

The area of rectangle is equal to

[tex]A=xy[/tex] -----> equation B

substitute equation A in equation B

[tex]A=x(60-x)[/tex]

[tex]A=-x^{2} +60x[/tex]

This is a vertical parabola open downward

The vertex is a maximum

The y-coordinate of the vertex of the graph is the maximum area of the garden and the x-coordinate is the length for the maximum area

using a graphing tool

The vertex is the point [tex](30,900)[/tex]

see the attached figure

Find the value of y

[tex]y=60-x[/tex] -----> [tex]y=60-30=30\ ft[/tex]

The dimensions of the rectangular garden is [tex]30\ ft[/tex] by [tex]30\ ft[/tex]

For a maximum area the garden is a square

The maximum area is [tex]900\ ft^{2} [/tex]

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