Question

The diagram shows a rectangular field with its length and breadth as indicated. Given that the perimeter of the field is 340 m, find the area of the field.

Answer 1

A = 6600 m²

Explanation:

In a rectangle

• opposite sides are congruent (equal ), then

3x + y = 100 - 4x ( add 4x to both sides )

7x + y = 100 ( subtract 7x from both sides )

y = 100 - 7x → (1)

Now we can express the length in terms of x only

2x + 3y = 2x + 3(100 - 7x) = 2x + 300 - 21x = 300 - 19x

the perimeter (P) of a rectangle is calculated as

P = 2( length + breadth)

given P = 340 , then

2(300 - 19x + 100 - 4x) = 340

2(400 - 23x) = 340 ( divide both sides by 2 )

400 - 23x = 170 ( subtract 400 from both sides )

- 23x = - 230 ( divide both sides by - 23 )

x = 10

substitute x = 10 into (1)

y = 100 - 7x = 100 - 7(10) = 100 - 70 = 30

Thus x = 10 and y = 30

Then

length = 2x + 3y = 2(10) + 3(30) = 20 + 90 = 110 m

breadth = 100 - 4x = 100 - 4(10) = 100 - 40 = 60 m

the area (A) of a rectangle is calculated as

A = length × breadth

= 110 × 60

= 6600 m²

Answer 2

• The length of the rectangular field is 100 m + (4x * 2) m = 100 m + 8x m.

• The breadth of the rectangular field is 6y m + (3x * 2) m = 6y m + 6x m.

Explanation:

Based on the diagram, we are given the following information:

The length of the rectangular field is 100 m + (4x * 2) m = 100 m + 8x m.

The breadth of the rectangular field is 6y m + (3x * 2) m = 6y m + 6x m.

We are also given that the perimeter of the field is 340 m. The perimeter of a rectangle is twice the sum of its length and breadth. Therefore, we can write the following equation:

2(100 m + 8x m + 6y m + 6x m) = 340 m

Simplifying the left-hand side of the equation, we get:

22x + 12y + 200 m = 340 m

Subtracting 200 m from both sides of the equation, we get:

22x + 12y = 140 m

Dividing both sides of the equation by 2, we get:

11x + 6y = 70 m

This is the equation that we will use to find the area of the field.

Finding the area of the field

The area of a rectangle is given by the formula:

Area = length × breadth

We need to find the values of length and breadth in terms of x and y. We can use the equation 11x + 6y = 70 m to solve for x or y.

Let's solve for x:

x = (70 m - 6y) / 11

Substituting this expression for x into the equation for the area, we get:

Area = (100 m + 8x m) × (6y m + 6x m)

Area = (100 m + 8((70 m - 6y) / 11) m) × (6y m + 6((70 m - 6y) / 11) m)

Expanding the parentheses, we get:

Area = (100 m + 560 m - 48y / 11) × (6y m + 420 m - 36y / 11)

Area = (660 m - 48y / 11) × (66y m - 36y / 11)

Multiplying the two factors, we get:

Area = 435600 m^2 - (2880y + 1728y^2) / 121

Area = 435600 m^2 - (4608y^2 + 2880y) / 121

This is the expression for the area of the field in terms of x and y.

Conclusion:

The area of the rectangular field can be expressed as:

Area = 435600 m^2 - (4608y^2 + 2880y) / 121

We can find the exact area of the field by substituting the value of y into the equation.