Solution:
Given:
Area of a rectangular land = 3000 sq m
and perimeter of the land = 220 m
To find:
Percentage of length or breadth to be decreased to make the square land.
Let, length (l) = x m and breadth (b) = y m
Then, by formula, area of rectangle = l x b i.e., xy = 3000 ------- ( equation 1)
and , perimeter of a rectangle = 2(l + b) = 2 ( x + y) = 220 i.e., x + y = 110 -------- (equation 2)
Substituting the value of y = 110 - x from equation 2 in equation 1 we get,
x (110 - x) = 3000
Or, 110x - x2 = 3000
Or, x2 - 110x + 3000 = 0
Or, x2 - (60x + 50x) + 3000 = 0
Or, x2 - 60x -50x + 3000 = 0
Or, x(x - 60) - 50(x-60) = 0
Or, (x - 50)(x - 60) = 0
Either, x -60 = 0 ----- (equation 3)
Or, x - 50 =0 --------- (equation 4)
From, eqn (3), x - 60 = 0 => x = 60
Now, putting the value of 'x' in equation 2 we get
y = 110 - 60 = 50
∴ If x = 60, then y = 50
Again, from equation 4
x - 50 =0 => x = 50. Putting x in eqn (2) we get, y = 110 - x = 110 - 50 = 60
∴ If x = 50, then y = 60
But assuming that length is longer than breadth, we get length(l) = 60 m and breadth (b) = 50 m
∴ Difference between length and breadth = 60 m - 50 m = 10 m
So, to make the land square the length has to be decreased by 10 m.
Here, Percentage decreased in length = \(\frac{difference}{length}\cdot100\%\)
= \(\frac{10m}{60m}\cdot100\%\)
= \(\frac{50}{3}\%\)
= 16.66%
∴ To change the given rectangular land into a square, length has to be decreased by 16.66%