a Consider that the length and width have to add to half the perimeter. Choose rectangles that have l + w = 200.
B
b Put the length and width of the rectangles from Part A in the table. Then multiply the dimensions to determine the area.
C
c If the y-coordinate is the area and the length is the x -coordinate, then the width can be represented by 200-x.
D
d Look for the highest point from the curve in Part C.
A
a See solution.
B
b See solution.
C
c See solution.
D
dLength: 100 feet Width: 100 feet
Practice makes perfect
a This exercise gives us a lot of freedom, there are literally thousands of rectangles that have a perimeter of 400 feet. Let's explore a series of them. It can be more efficient to get the length and width by thinking about the perimeter of half of the rectangle.
cc
P&=&2l+2w
&⇕
P/2&= &2l+2w/2
&⇕
400/2&=&l +w
In our case, the perimeter is 400 meaning half of that would be 200. We can choose numbers for our length and width that add to 200. Let's draw some rectangles.
Keep in mind those are just four possible solutions.
b We need to find the area of each rectangle from Part A, plus an additional rectangle. Let's put it in the suggested table. Recall that length and width add up to 200.
Rectangle
Length
Width
Area
1
160 ft
40 ft
6400 ft^2
2
150 ft
50 ft
7500 ft^2
3
120 ft
80 ft
9600 ft^2
4
x ft
200-x ft
x(200-x) ft^2
c For this part, we need to make a graph that relates the width to the area. Let's add a row to our table from Part B and get some more data points to see if there is pattern.
Length (x)
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
x
Width
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
200-x
Area (y)
1900
3600
5100
6400
7500
8400
9100
9600
9900
10 000
9900
9600
9100
8400
7500
6400
5100
3600
1900
200x-x^2
Now let's look at the graph of the equation y=200x-x^2.
The peak of the graph is the largest possible area, and that is when y = 10 000.
d We can use the largest possible area from Part C, 10 000 ft^2 and use the table to determine the length and width that gave us that area.
Length
50
60
70
80
90
100
Width
150
140
130
120
110
100
Area
7500
8400
9100
9600
9900
10 000
This table indicates that both the length and the width are 100 ft when the area is at its maximum.
