Find the length of the boundary of the square field enclosing the lawn and the path

Question

ABCD is a square lawn of area 729 metre square and a path of uniform width runs all around it , area of which is 295 metre square.Find the length of the boundary of the square field enclosing the lawn and the path, and also the width of the path.

Answer 1

If ABCD is square, with side length x and a path of width y around the square, then the area of the path is:

4xy (the parts of the path bordering the square)

4y^2 (the square bits touching the corners of square ABCD).

The area of the path is 4xy + 4y^2

Square ABCD, with an area of 729, has a side length of 27, so x = 27.

The area of the path is given as 295, so:

4xy + 4y^2 = 295

4(27)y + 4y^2 = 295

108y + 4y^2 = 295

4y^2 + 108y - 295 = 0

Quadratic formula. . .

y = (-108 +- sqrt( 108^2 - 4(4)(-295) ) ) / 2(4)

y = (-108 +- sqrt(11664 + 4720) ) / 8

y = (-108 +- sqrt(16384) ) / 8

y = (-108 +- 128) / 8

It's a physical measurement, so it can't be negative. . .

y = (-108 + 128) / 8

y = 20/8

y = 2.5

The width of the path is y = 2.5, but what about the outer square (with the path surrounding the inner square)?

The outer square is x + 2y because the path sticks out on both sides of the inner square.

27 + 2(2.5) = 27 + 5 = 32

Answer:  The path is 2.5 meters wide, the inner square is 27 meters wide, and the outer square is 32 meters wide.