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