a farmer has a 2040 ft of fencing and wishes to fence off two separate fields. one of the fields is to be a rectangle with the length twise as long os the width. while the other field is to be square. determine the dimension of the fields if the farmer wishes to maximize the total area of the two fields.
the diagram has a rectangle with the width=4 and the length=2x
the total perimeters =2040.
To determine the dimensions of the fields that will maximize the total area, we need to set up equations using the given information.
Let's start with the rectangular field. It is given that the length is twice as long as the width. So, if the width is represented by "w," then the length would be "2w." The perimeter of a rectangle is calculated by adding all four sides:
Perimeter = 2(length + width)
For the rectangular field, we have:
2(2w + w) = 2040
Simplifying this equation, we get:
6w = 2040
w = 340
So, the width of the rectangular field is 340 ft. Since the length is twice the width, the length would be 2 * 340 = 680 ft.
Next, let's move on to the square field. The perimeter of a square is calculated by multiplying the length of one side by 4:
Perimeter = 4 * side length
Since all sides of a square are equal, we can represent the side length as "s." Therefore, the equation for the square field is:
4s = 2040
Simplifying this equation, we get:
s = 510
So, the side length of the square field is 510 ft.
Now, to determine the dimensions that will maximize the total area, we need to calculate the area of each field and add them together.
The area of the rectangular field is calculated as:
Area = length * width
Area = 680 * 340 = 231,200 sq ft
The area of the square field is calculated as:
Area = side length * side length
Area = 510 * 510 = 260,100 sq ft
The total area is the sum of the areas of the two fields:
Total Area = Area of Rectangular Field + Area of Square Field
= 231,200 sq ft + 260,100 sq ft
= 491,300 sq ft
Therefore, the dimensions that will maximize the total area are as follows:
The rectangular field has a width of 340 ft and a length of 680 ft.
The square field has a side length of 510 ft.