Learning Goal
By the end of this lesson I will be able to determine the maximum area for a rectangle if given the perimeter.
Optimization
The area is the space contained within a two dimensional object. Optimization is the process of making something as efficient as possible by selecting the best available alternative. In order to optimize an area it is important to find the shape with the maximum area.
Why the Maximum Area?
Let's pretend that you own a zoo and are about to create a new exhibit for the zebra's but can only afford to use 100m worth of fencing. There are many different pens that can be created and they will result in varying areas inside of them.
Fencing is expensive, but the quality of the animal's lives are also important so you want to create a rectangular pen that has a lot of room inside. A long and skinny pen does not give the zebra enough room to roam comfortably, so the best alternative in this situation is to create the pen with the largest area.
Example A. A splash pad builder wants to enclose the maximum area possible for fun, but only has enough materials for a perimeter of 40m. Determine the maximum area possible.
One way to determine maximum area is to use a chart and write out some of the possible values that will give a perimeter of 40m. The length and width, when combined using the formula \begin{align*}Perimeter = 2(length) + 2(width)\end{align*} must equal 40.
Length (m) | Width (m) | Perimeter (m) | Area (m^{2}) |
1 | 19 | 40 | 19 |
2 | 18 | 40 | 36 |
4 | 16 | 40 | 64 |
5 | 15 | 40 | 75 |
8 | 12 | 40 | 96 |
10 | 10 | 40 | 100 |
12 | 8 | 40 | 96 |
The area column can be completed by using the formula for the area of a rectangle, \begin{align*}Area = (length)(width)\end{align*}. The rectangle with the largest area has dimensions 10m by 10m. The pen with the maximum area is square in shape.
A Square is the Best
In general, the most efficient rectangle is a square. The rectangle with the largest area for a given perimeter will always be a square. This means that, given a perimeter, the optimal rectangle will have a length and a width that are the same size. In other words, to find the length (or width) of the rectangle with the optimal dimensions, one just has to divide the total perimeter by four.
Example B. Julie is constructing a rectangular patio in her backyard with a railing around the outside. Julie can only afford to pay for 48m of railing. What size patio would produce teh maximum area for entertaining?
What you know is that the perimeter is 48m. The optimal rectangle is a square, so the length and width are both the same size. This is indicated below as \begin{align*}4l\end{align*}.
\begin{align*}Perimeter = l + w + l +w\end{align*}
\begin{align*}Perimeter = l + l + l + l\end{align*}
\begin{align*}48 = 4l\end{align*}
\begin{align*}4 = l\end{align*}
Therefore the patio with the maximum area has dimensions of 4m by 4m. The maximum area would be
\begin{align*}Area = (length)(width)\end{align*}
\begin{align*}Area = (4)(4)\end{align*}
\begin{align*}Area = 36m^2\end{align*}
Three Sides Instead of Four
Example C. A rancher is building a rectangular pen beside a river. The river will form one side of the rectangle as shown in the diagram below. The rancher wants the pen to have a perimeter of 800m. What dimensions will give the largest area?
Note the fact that this rectangular pen only has three sides. It has one length measurement and two width measurements. Using a table once more,
Length (m) | Width (m) | Perimeter (m) | Area (m^{2}) |
700 | 50 | 800 | 35 000 |
600 | 100 | 800 | 60 000 |
500 | 150 | 800 | 75 000 |
400 | 200 | 800 | 80 000 |
200 | 300 | 800 | 60 000 |
The maximum area occurs when the length is 400m and the width is 200m. The square does not have the maximum area, because the pen only has three sides.
For a review of the concept covered, please watch the following video.
Practice
Directions: Given the following rectangles, determine the maximum area for the given perimeter.
1. A rectangle with area 80m^{2 }(solution: 400m)
2.A rectangle with area 44 m^{2 } (solution: 121m)
3. A rectangle with area 35 cm^{2} (solution: 76.6cm)
4. A farmer constructs a rectangular corral on the side of a barn. The barn will make up one full side of the pen. The perimeter of fence on the other three sides totals 400m. Determine the dimensions that give a maximum area and determine this area.
(solution: 100m x 200m with a perimeter of 20 000m)