Construction Your community is building a park. It wants to fence in a play space for toddlers. It wants the maximum area for a given amount of fencing. Which shape, a square or a circle, provides a more efficient use of fencing?
One measure of efficiency is the ratio of area fenced to fencing used, or area to perimeter. Which of the two shapes has the greater ratio?
Square | Circle | |
---|---|---|
|
Define area and perimeter. |
|
|
|
|
|
Express s and r in terms of a common variable, P. |
|
|
Write the ratios. Substitute for s and r. Simplify. |
|
How can you compare
Since the numerators are the same, the fraction with the smaller denominator is the larger fraction.
Since
Check Assume
The area of the square is
Simplify each rational expression. State any restrictions on the variables.
Multiply or divide. State any restrictions on the variables.
Reasoning The width of the rectangle is