The valid dimensions of the football pitch are approximately:
Length ≈ 118.24 meters
Breadth ≈ 88.68 meters
To find the dimensions of the football pitch, where the area is 7140 square meters and the length and breadth are represented by the expressions:
Length = 4x + 5
Breadth = 3x - 7
We need to set up an equation based on the area of the football pitch and then solve for 'x'. Once we find the value of 'x', we can use it to calculate the length and breadth. Here are the detailed steps:
Step 1: Set up the equation based on the area:
Area = Length × Breadth
Given that the area is 7140 square meters, we have:
7140 = (4x + 5) × (3x - 7)
Step 2: Expand and simplify the equation:
7140 = 12x^2 - 28x + 15x - 35
Combine like terms:
7140 = 12x^2 - 13x - 35
Step 3: Move all terms to one side of the equation:
12x^2 - 13x - 35 - 7140 = 0
Step 4: Simplify the equation:
12x^2 - 13x - 7175 = 0
Now, we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 12, b = -13, and c = -7175. We'll solve this quadratic equation for 'x' using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values:
x = (-(-13) ± √((-13)² - 4 * 12 * (-7175))) / (2 * 12)
x = (13 ± √(169 + 344400)) / 24
x = (13 ± √344569) / 24
Step 5: Calculate the values of 'x':
We have two possible values for 'x' based on the ± sign:
1. x₁ = (13 + √344569) / 24
2. x₂ = (13 - √344569) / 24
Now, calculate these values:
1. x₁ ≈ 29.56 (rounded to two decimal places)
2. x₂ ≈ -94.39 (rounded to two decimal places)
Step 6: Calculate the dimensions of the football pitch using these values of 'x':
Length = 4x + 5
Breadth = 3x - 7
For x₁ ≈ 29.56:
Length₁ ≈ 4(29.56) + 5 ≈ 118.24 meters
Breadth₁ ≈ 3(29.56) - 7 ≈ 88.68 meters
For x₂ ≈ -94.39:
Length₂ ≈ 4(-94.39) + 5 ≈ -377.56 meters (Negative length doesn't make sense in this context)
Breadth₂ ≈ 3(-94.39) - 7 ≈ -290.17 meters (Negative breadth doesn't make sense in this context)
So, the valid dimensions of the football pitch are approximately:
Length ≈ 118.24 meters
Breadth ≈ 88.68 meters