7.1.2 The Relationship between Length and Area
7.1.2 • The Relationship between Length and Area
Activity
Noah has 50 meters of fencing to completely enclose a rectangular garden in the backyard.
Draw some possible diagrams of Noah’s garden. Label the length and width of each rectangle.
Your answers will vary. Did you consider organizing your lengths, widths, and areas into a table? For example:
| Length (meters) | Width (meters) | Area (square meters) |
|---|---|---|
| 5 | 20 | 100 |
| 10 | 15 | 150 |
| 12 | 13 | 156 |
| 12.5 | 12.5 | 156.25 |
| 18 | 7 | 126 |
| 20 | 5 | 100 |
| 24 | 1 | 24 |
Find the length and width of a rectangle that would produce the largest possible area. Explain why you think that pair of length and width gives the largest possible area.
Compare your answer:
When the length and width are each 12.5 meters, the area is the largest (156.25 square meters). A table shows that as the rectangular shape gets closer and closer to being a square, the area gets larger.
Self Check
Additional Resources
Largest Possible Area
Tyrel is enclosing a grassy area for his goats. He has 80 feet of fencing and wants to make a rectangular enclosed area. What are the dimensions of the enclosure with the largest possible area?
First, make a table with possible length and width dimensions and their areas. Remember, each and .
| Length (meters) | Width (meters) | Area (square meters) |
|---|---|---|
| 5 | 35 | 175 |
| 10 | 30 | 300 |
| 15 | 25 | 375 |
| 20 | 20 | 400 |
| 25 | 15 | 375 |
| 30 | 10 | 300 |
| 35 | 5 | 175 |
Notice that the closer the rectangle is to a square, the larger the area.
The largest area occurs when the length and width are both 20 feet for an area of 400 square feet.
Try it
Largest Possible Area
Elena has 100 feet of fencing to make a rectangular pen for her cow. What is the largest possible area she can make with this amount of fencing?
Here is how to find the largest possible area:
Make a table with possible dimensions:
| Length (meters) | Width (meters) | Area (square meters) |
|---|---|---|
| 10 | 40 | 400 |
| 20 | 30 | 600 |
| 25 | 25 | 625 |
| 30 | 20 | 600 |
| 40 | 10 | 400 |
Notice the closer you are to a square, the larger the area.
The largest possible area is 625 square feet.