A gardener has a rectangular plot of land bordered on one side by a brick wall. If she
has 32 feet of edging, what is the maximum area that can be enclosed? What are the dimensions
of the plot that yield this area?
To find the maximum area enclosed by the rectangular plot, we can use the concept of optimization. Let's assume the length of the plot is 'l' and the width is 'w'.
The perimeter of the rectangular plot is given as 32 feet:
Perimeter = 2l + w = 32
We need to express w in terms of l:
w = 32 - 2l
Now, the area of the rectangular plot is given by the product of its length and width:
Area = l * w
Substituting the value of w, we have:
Area = l * (32 - 2l)
To find the maximum area, we need to determine the value of 'l' that maximizes the function 'Area'. We can use calculus to do this.
First, let's expand the function:
Area = 32l - 2l^2
To find the maximum or minimum of a function, we take its derivative and set it equal to zero:
d(Area)/dl = 32 - 4l
Setting it equal to zero:
32 - 4l = 0
Solving for 'l':
4l = 32
l = 8
Now that we have the value of 'l', we can find the corresponding value of 'w':
w = 32 - 2l = 32 - 2*8 = 32 - 16 = 16
So, the dimensions of the plot that yield the maximum area are:
Length (l) = 8 feet
Width (w) = 16 feet
To obtain the maximum area, the gardener should create a rectangular plot with dimensions 8 feet by 16 feet, enclosing an area of 128 square feet.