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प्रश्न
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?
उत्तर
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
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