Question

A garden measuring 12m by 16m is to have a pedestrian pathway that is ‘ω’ meters wide installed all the way around so that it increases the total area to 285 m². What is the width of the pathway?

Answer 1

Let the width of the rectangle be “ω”

Length of the outer rectangle = 16 + (ω + ω)

16 + 2ω

Breadth of the outer rectangle = 12 + 2ω

By the given condition

(16 + 2ω) (12 + 2ω) = 285

192 + 32ω + 24ω + 4ω² = 285

4ω² + 56ω = 285 – 192

4ω² + 56 ω = 93

4ω²+ 56 ω – 93 = 0

Here a = 4, b = 56, c = – 93

x = `(-"b" ± sqrt("b"^2 - 4"ac"))/(2"a")`

ω = `(- 56 ± sqrt(56^2 - 4(4)(-93)))/8`

= `(- 56 ± sqrt(3136 + 1488))/8`

= `(- 56 ± sqrt(4624))/8`

= `(- 56 ± 68)/8`

ω = `(- 56 + 68)/8` or `(- 56 - 68)/8`

= `12/8` or `-124/8`

= 1.5 or – 15.5 ...(Width is not negative)

∴ Width of the path way = 1.5 m