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A gardener decides to lay a stone pathway around the outside of a rectangular garden. If he has 220 ft of stone what is the dimension of the garden that provides the maximum area using these stones.

a) Draw a sketch to represent this situation. Include appropriate expressions in single variable for each side. Show works for how you got the expressions for the length and width.

b.) Find the dimensions of the garden for its maximum area using the 220 ft stone. Draw a graph that would represent the equation you used to find the maximum area. Show work. State your answer clearly list the maximum area and dimensions.

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  • $\begingroup$ can you show us your work? What dont you understand? $\endgroup$ – rannoudanames Oct 2 '16 at 18:18
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The maximum area occurs when the rectangle has the dimensions of a square. If the gardener has 220 ft of stone to work with, then the dimensions will be:

220/4 = 55 ft x 55 ft = max dimensions

If you wanted to find the max area, just multiply 55 x 55 and you get 3025 ft^2

Now, the reason it is 55 is because:

Perimeter = 2 (l + w)

220 = 2 (l + w)

110 = l + w

110 - w = l

Area = lw

Area = (110 - w)(w)

This equation above is a quadratic equation. In order to find the maximum area, we can find the midpoint of the zeroes (also known as the x-coordinate of the vertex)

x1 = 110 and x2 = 0

(110 + 0)/2 = 55 = w

110 = l + w

110 = l + 55

55 = l

Therefore, it is 55 ft x 55 ft

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