A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 meters, assuming the width to be x, form an equation in x and solve it to find the value of x.
First let us see what the question is demanding and then provide as per the demand of the question we will proceed step wise to get the exact answer.
Given –
The length and width of rectangle = 10 and 16
from which we can find the area of rectangle
formula for area of rectangle = Length x width
As per the question
Area of the rectangle = 16 x 10 = 160 m ( we will call this outer rectangle).
Area of other rectangle after the construction of wall = 120 (given in question. we will call this inner rectangle).
Demand of question is to form and equation in “x” and then solve for “x”
area of inner rectangle = area of outer rectangle (walk+ wall) – area of inner rectangle(walk only)
120 (given in question) = 10 *16 – (16+2x) * (10+2x)
120 = 160 – [10(16+2x)+2x(16+2x)]
120=160 – [10*16+10*2x + 2x*16+2x*2x]
120=160 – [160 + 20x + 32x + 4 x² ]
120 = 160 – [160 + 52x +4 x² ]
120 = 160 – 160 -52x – 4 x²
120 = -52x -4 x²
4 x² +52x -120 = 0
Now solve for x
4 x² + 52x – 120 = 0
Dividing the whole equation by 4, we get
x² + 13 x – 30 = 0
now, using trial and error method we will find the best combination,
we have to find such nos.which on addition or subtraction produces 13 and the same nos. if multiplied produce 30
nos. will be 15-23 = 13 and 15*2=30
putting these in above equation we get,
x² + 15x -2x – 30 = 0
x(x+15) – 2 (x + 15) = 0
(x+15) and (x-2) = 0
x=-15 and x = 2
-15 is not acceptable hence x= 2 is the answer.