A rectangular field is 140 yards long and 60 yards wide. Give the length and width of angrier rectangular field that has the same perimeter but a larger area
clearly, if you increase both the length and the width, both the area and the perimeter will increase.
Maximum area for a given perimeter is achieved for a square.
So, 100x100 still has the same perimeter of 400, but larger area.
Other choices could be
139x61
138x62
...
102x98
101x99
You are correct, increasing both the length and width will result in both the area and the perimeter increasing. Additionally, a square will have the maximum area for a given perimeter.
So, for a rectangle with a perimeter of 400, the largest area will be achieved with a square. In this case, the side length of the square would be 100 yards, resulting in an area of 100 * 100 = 10,000 square yards.
Other rectangular options with the same perimeter of 400 include:
- A rectangle with dimensions 139 yards by 61 yards, resulting in an area of 139 * 61 = 8,479 square yards.
- A rectangle with dimensions 138 yards by 62 yards, resulting in an area of 138 * 62 = 8,556 square yards.
- and so on...
The area will continue to increase as the difference between the length and width decreases. Another example includes a rectangle with dimensions 101 yards by 99 yards, resulting in an area of 101 * 99 = 9,999 square yards.
To find the length and width of a larger rectangular field with the same perimeter but a larger area, we need to understand the relationship between perimeter, length, and width.
First, recall that the perimeter of a rectangle is found by adding the lengths of all four sides. In this case, the perimeter of the given rectangular field is calculated as:
Perimeter = 2(Length + Width)
Given that the length of the rectangular field is 140 yards and the width is 60 yards, we can substitute these values into the perimeter formula:
Perimeter = 2(140 + 60)
Perimeter = 2(200)
Perimeter = 400 yards
Since we want to find another rectangular field with the same perimeter but a larger area, but we don't know the dimensions, we can use variables. Let's assume the new length is L2 and the new width is W2.
Now, we can construct an equation to relate the perimeter and the area of both rectangular fields:
Perimeter1 = Perimeter2
2(Length1 + Width1) = 2(Length2 + Width2)
Area1 < Area2
Length1 * Width1 < Length2 * Width2
Since the perimeters of both fields are equal, we can equate the expressions for the perimeter:
2(Length1 + Width1) = 2(Length2 + Width2)
Since we want to find the dimensions of the new field with a larger area, we want to maximize the product (Length2 * Width2), given the constraint (2(Length1 + Width1) = 400).
To solve this problem, we need to use a little trial and error. We can start by substituting values that satisfy the perimeter equation and check if the area increases.
Let's try increasing the length and reducing the width.
Let's assume the new length (L2) is 180 yards and the new width (W2) is 50 yards.
Plugging in these values into the perimeter equation:
2(140 + 60) = 2(180 + 50)
400 = 460
The perimeter equation is not satisfied, meaning that the dimensions are not correct.
We need to try a different combination. Let's decrease the length and increase the width.
Assuming the new length (L2) is 120 yards and the new width (W2) is 80 yards.
Plugging in these values into the perimeter equation:
2(140 + 60) = 2(120 + 80)
400 = 400
The perimeter equation is satisfied.
Now, let's check if the area has increased:
Area1 = Length1 * Width1 = 140 * 60 = 8400 square yards
Area2 = Length2 * Width2 = 120 * 80 = 9600 square yards
As we can see, the area of the second rectangular field (9600 square yards) is greater than the area of the first rectangular field (8400 square yards).
Therefore, the length and width of the new rectangular field, which has the same perimeter but a larger area, are 120 yards and 80 yards, respectively.
To find a rectangular field with the same perimeter but a larger area, we need to increase either the length or the width or both.
Let's start by increasing the length of the field. If we increase the length, we will also need to increase the width to maintain the same perimeter.
Let's assume we increase the length by x yards. This means the new length will be 140 + x yards. To maintain the same perimeter, we also need to increase the width by x yards. This means the new width will be 60 + x yards.
The perimeter of the original rectangular field is 2(length + width) = 2(140 + 60) = 400 yards.
So, the perimeter of the new rectangular field is 2((140 + x) + (60 + x)) = 400 yards.
Simplifying the equation, we get:
2(200 + 2x) = 400
400 + 4x = 400
4x = 0
x = 0
From this, we can conclude that if we only increase the length of the field, the area will remain the same. Therefore, to find a rectangular field with the same perimeter but a larger area, we need to increase both the length and the width.
However, since we don't have any restrictions on the increase in length and width, we can choose any values to increase them.