Answer :
A rectangular field with a width of 100 meters and a length of 100 meters would have the same perimeter as the given field (80 meters wide and 120 meters long) but a larger area.
To solve this problem, we need to find another rectangular field with the same perimeter but a larger area compared to the given field. Let's go step by step:
1. Find the perimeter of the given field:
Perimeter = 2 * (Length + Width)
= 2 * (120m + 80m)
= 2 * 200m
= 400m
2. Determine the area of the given field:
Area = Length * Width
= 120m * 80m
= 9600m²
3. Let's assume the length of the new rectangular field is x meters. Since both fields have the same perimeter, the new field's width can be calculated using the formula for the perimeter:
Perimeter = 2 * (Length + Width)
400m = 2 * (x + Width)
200m = x + Width
4. Now, we need to express the width in terms of x:
Width = 200m - x
5. The area of the new rectangular field can be calculated using the width and length:
Area = Length * Width
= x * (200m - x)
6. To find the dimensions that yield the largest area, we need to find the maximum point of the area function. Let's take the derivative of the area function with respect to x and set it equal to zero:
d(Area)/dx = 0
d(x * (200m - x))/dx = 0
200m - 2x = 0
2x = 200m
x = 100m
7. We substitute the value of x back into the equation for the width:
Width = 200m - x
= 200m - 100m
= 100m
8. Therefore, the length and width of the new rectangular field with the same perimeter but a larger area are:
Width = 100 meters
Length = 100 meters
In summary, a rectangular field with a width of 100 meters and a length of 100 meters would have the same perimeter as the given field (80 meters wide and 120 meters long) but a larger area.
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