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Mr. Sanchez has 16 feet of fencing to put around a rectangular garden. He wants the garden to have the greatest possible area. How long should the sides of the garden be?

Sagot :

jclainwizard
to create a garden w/ the largest area you want to make a garden that is a square or as close as possible 16 is a perfect square # so the sides should be 4 ft each

Answer:

The sides  of the garden be 4 feet to have the greatest possible area.

Step-by-step explanation:

Given :   Mr. Sanchez has 16 feet of fencing to put around a rectangular garden. He wants the garden to have the greatest possible area.

We have to determine the sides of the garden so that the garden to have the greatest possible area.

Let Length of garden be x feet

and width of garden be y feet.

Then given perimeter = 16 feets.

Perimeter of rectangular garden = 2(length + width)

⇒ 2x + 2y = 16

⇒ x + y = 8

⇒  y = 8 - x  

Thus, area of rectangle is = Length  × width

A = x × y

A = x (8-x) = [tex]8x-x^2[/tex]

For area to be maximum applying derivative test

Differentiate [tex]A=8x-x^2[/tex] with respect to x, we have,

[tex]\frac{dA}{dx}=8-2x[/tex]

Put [tex]\frac{dA}{dx}=0[/tex] , we have,

[tex]8-2x=0 \Rightarrow 8=2x \Rightarrow x=4 [/tex]

Thus, y = 8- x = 8 - 4 = 4

Thus, The sides  of the garden be 4 feet to have the greatest possible area.

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