A farmer bought 16 meters of fencing. He wants to build a rectangular area for his baby chicks. How could the farmer build a fenced in area using all of the fencing, but creating the largest area for his chicks?
Using graph paper, cut out, glue and label all the possible solutions. Clearly label the perimeter and area of each shape.
a 4x4 square
To find the largest area for the rectangular fenced area, we need to consider different dimensions that can maximize the area while using the given 16 meters of fencing.
Let's start by understanding the perimeter of a rectangle. Perimeter is the total distance around the outside of a shape, in this case, the fenced area. For a rectangle, the perimeter is calculated by adding the lengths of all the sides.
In our case, the farmer has 16 meters of fencing, so the perimeter of the rectangle should be 16 meters.
To maximize the area, we need to find the dimensions (length and width) that can satisfy the given perimeter while providing the largest possible area.
One way to approach this is to create a table and list down different combinations of length and width that add up to 16 meters. We can start with a length of 1 meter and find the corresponding width.
Length (L) | Width (W) | Perimeter (2L + 2W)
-------------------------------------------
1 | 7 | 16
2 | 6 | 16
3 | 5 | 16
4 | 4 | 16
5 | 3 | 16
6 | 2 | 16
7 | 1 | 16
From the table, we see that when the length is 1 meter, the width is 7 meters, resulting in a perimeter of 16 meters. Similarly, when the length is 2 meters, the width is 6 meters, and so on.
Next, we calculate the area by multiplying the length and width of each combination:
Length (L) | Width (W) | Perimeter (2L + 2W) | Area (L * W)
---------------------------------------------------------
1 | 7 | 16 | 7
2 | 6 | 16 | 12
3 | 5 | 16 | 15
4 | 4 | 16 | 16
5 | 3 | 16 | 15
6 | 2 | 16 | 12
7 | 1 | 16 | 7
Now, we can see the area for each combination.
Using graph paper, cut out rectangles with the corresponding dimensions from the above table. Label each shape with its perimeter and area. Arrange the rectangles from the largest area to the smallest area.
The rectangles in order of the largest area to the smallest area are:
- Rectangle with length 4 meters and width 4 meters (Perimeter: 16 meters, Area: 16 square meters)
- Rectangles with length 3 meters and width 5 meters, and length 5 meters and width 3 meters (Perimeter: 16 meters, Area: 15 square meters)
- Rectangles with length 2 meters and width 6 meters, and length 6 meters and width 2 meters (Perimeter: 16 meters, Area: 12 square meters)
- Rectangles with length 1 meter and width 7 meters, and length 7 meters and width 1 meter (Perimeter: 16 meters, Area: 7 square meters)
By arranging the rectangles in this order, you have visually represented the solution to maximize the area within the given 16 meters of fencing for your baby chicks' fenced area.