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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

 

Solution:

 

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!

 

 

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