a garden measuring 12meters by 16meters, path total area 285square ft what will be width

Question

A garden measuring 12 meter by 16 meters is to have a pedestrian pathway installed all around it increasing the toal are to 285 square meters. What will the width be

Answer 1

The path is x meters wide.

The garden is 12 meters wide by 16 meters long.

The perimeter of the garden is 56 meters.

The area of the path is 56 * x + 4*x^2.

The 56x is the part of the path bordering the garden.

The 4x^2 is the corners of the path diagonal to the garden's corners.

The area of the garden is 216 meters^2.

The area of the path is 285 - 216 = 69 meters^2

56x + 4x^2 = 69

4x^2 + 56x - 69 = 0

Quadratic

x = (-56 +- sqrt(56^2 - 4(4)(-69))) / 2(4)

x = (-56 +- sqrt(4240)) / 8

x = (-56 +- sqrt( 16 * 265)) / 8

x = (-56 +- 4sqrt(265)) / 8

The +- has to be + since we can't have a walkway with negative width.

x = (-56 + 4sqrt(265)) / 8

x = about 1.14

Answer:  About 1.14 meters wide.

Answer 2

The garden area is 12*16=192 sq m.

The path forms a larger rectangle around the garden rectangle, and its sides have lengths 16+2w and 12+2w, with an area of (16+2w)(12+w) [draw a picture; it will help].

Expanding the brackets: 192+32w+24w+4w^2 [this is where 4x^2 comes in in the other answer]=285, the total area of garden and path. So we can write this: 4w^2+56w-93=0. 93 is the difference between the garden area and the total area with path.
Hopefully this factorises: (2w-3)(2w+31)=0. Obviously w must be positive for our purposes so we take the solution 2w=3 metres, making w=1.5m.
Let's check it out: width of wider garden=15 and length=19. Area=19*15=285 sq m.