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If farmer is enclosing a rectangular garden, using one side of his bone, the farmer has 80 m of fencing for the remaining three sides the length will be longer than the width. This diagram shows a bone of the garden width in meters result in the largest possible area for the rectangular garden.

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The functions and dimensions of the rectangular garden are;

a. A = l·(80 - l)/2

b. 800 square meters

c. Length; 40 meters

Width; 20 meters

d. Length; 150 meters

Width; 75 meters

The details of the steps used to find the above values are as follows;

The dimensions of the rectangular area, indicates that we get;

80 m = 2·w + l

Therefore; The width of the garden, w = (80 - l)/2

The area of a rectangle is; Length × Width

Area of the garden is; l × w = l × (80 - l)/2

A = l × (80 - l)/2

b. A = l × (80 - l)/2

The maximum ares can be obtained by equating the derivative for the function for area to zero as follows;

dA/dl = d(l × (80 - l)/2)/dl

d(l × (80 - l)/2)/dl = d((80·l - l²)/2)/dl

d((80·l - l²)/2)/dl = 40 - l

dA/dl = 40 - 1

The maximum area can be obtained at the point where we get;

dA/dl = 0

Therefore; 40 - l = 0

l = 40

The maximum area is; 40 × (80 - 40)/2 = 800 m²

c. The length of the garden that produces the maximum area is l = 40 m

The width, w = (80 - l)/2

The width of the garden that produces the maximum area, w is therefore;

w = ((80 - 40)/2) m

Therefore, w = 20 m

d. Where the length of the fencing available is 300 meters, we get;

d((300·l - l²)/2)/dl = 150 - l

The maximum area can be obtained when we get;

dA/dl = 0, therefore;

150 - l = 0

l = 150 meters at the maximum area

The width, w is; (300 - 150)/2 = 75 meters

The complete question, obtained from a similar question found through search can be presented as follows;

A farmer has 80 meters of fencing to use to enclose a rectangular garden that has one side against a barn

Let l represent the length of the rectangular garden (in meters) and let A represent the area of the rectangular garden (in square meters)

a. Write a formula that expresses A in terms of l

A = ___

b. What is the maximum possible area of the garden (It may help to use a graphing calculator)

___ Square meters

c. What is the length and width of the garden that produces the maximum area?

Length ___ meters

Width ___ meters

d. If the farmer instead had 300 meters of fencing to enclose the garden, what garden-length and garden-width will produce a garden with the maximum area?

Length ___ meters

Width ___ meters

If farmer is enclosing a rectangular garden, using one side of his bone, the farmer-example-1
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