Question

A rectangular garden 40 m ×30 m is surrounded from outside by a path of equal width. if the area of the path is 456 m² , find the width of the path?

Answer 1

QUESTION :

A rectangular garden 40 m ×30 m is surrounded from outside by a path of equal width. if the area of the path is 456 m² , find the width of the path?

ANSWER :

FOR RECTANGLE

LENGTH = 40M

BREADTH = 30M

SO AREA OF GARDEN=L×B

=40×30

=1200M²

SO LET THE SUM OF WIDTH BE X FOR THE PATH FROM BOTH SIDES

SO 2(WIDTH)=X

SO OVERALL LENGTH = (40+X)M

AND OVERALL BREADTH = (30+X)M

SO OVERALL AREA = L×B

=(40+X)(30+X)

=1200+40X+30X+X²

= (X²+70X+1200)M

NOW IT IS GIVEN THAT

AREA OF PATH=456M²

SO AREA OF PATH = OVERALL AREA-AREA OF GARDEN

456=X²+70X+1200-1200

X²+70X-456=0

So by splitting the middle term method

X²+76X-6X-456=0

X(X+76)-6(X+76)=0

(X+76)(X-6)=0

SO

X+76=0 OR X-6=0

X=-76 OR X=6

BUT BREADTH CAN NEVER BE NEGATIVE

SO X=-76 IS REJECTED

SO X=6 IS ACCEPTED

SO 2(WIDTH)=X

WIDTH=X/2

WIDTH=6/2

WIDTH=3M

HENCE THE WIDTH OF THE PATH IS 3 METRES

Answer 2

Mensuration

Dimension of rectangle $= 40\ m \times 30\ m$

As it is given there is  a path of equal width all around.

Area of path is $456\ m^2$.

Let the width of path be x.

Area of path $= area \ of\ total\ land\ including\ paths-area\ of\ initial\ rectangular\ plot$

$\implies [(40+x)\times (30+x)]-(40\times 30)=456\ m^2\\\\\implies (1200+40x+30x+x^2)-(1200)=456\\\\\implies x^2+70x=456\\\\\implies x^2+70x-456=0$

$\implies x^2+76x-6x-456=0\\\\\implies x(x+76)-6(x+76)=0\\\\\implies (x+76)(x-6)=0\\\\\implies x=-76 \ and \ x=6$

-76 can't b e the answer so, 6 is the answer.

Hence, $\frac{6}{2} =3\ m$  is the width of the path.