Area Calculation MCQ 3 (Path occupying an area runs inside a rectangle, to find width of the rectangle)
Area Calculation MCQ 3: A path of uniform width runs round the inside of a rectangular field 38 m long and 32 m wide. if the path occupies 600m2, then the width of the path is_______?
a) 3 m b) 5 m c) 8 m d) 10 m
We know that;
\[ Area of rectangle = Length \times Width \]
As per given length and width the are of the rectangle is ;
\( Area of The rectangle = 38\times32=1216 m^2 \)
From this area we subtract the area of the path;
==>\[ Area Of Small Rectangle Surrounded By The Path=1216-600=616 m^2 \]
i-e for inner rectangle \( Length \times Width = 616 \)
But what is the length and width of inner triangle ?
To find this we suppose that the path has width ‘x’
Then from fig we can see that
Length of smaller rectangle = (38-2x) m
Width of inner rectangle = (32-2x) m
So now we can write ;
\[ (38-2x)(32-2x)=616 \]
\[ 1216-76x-64x+4x^2=616 \]
\[ 4x^2-140x+600=0 \]
\[ x^2-35x+150=0 \]
\[ x^2-30x-5x+150=0 \]
\[ ==> (x-30)(x-5)=0 \]
==> x= 30 or x=5
If we choose x = 30 as our answer then as we set width of smaller rectangle to be (38-2x) – will become 38-2(30)=38-60=-22 which is wrong – because a length is never -ve !
So we choose x=5 m as our answer.
So option b is correct!