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QJEEM17Ma71c3: JEE Main 2017 Mathematics - Class 12 JEEM17Ma71c3 Solution: Twenty meters of wire is available for fencing off a flower-bed …

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

Multiple Choice Options

  • A 10
  • B 25
  • C 30
  • D 12.5

Step-by-Step Verified Solution

JEE Main 2017 (Offline) Mathematics - Application of Derivatives Question 173 English Explanation We have

Total length = r + r + r θ\theta = 20

⇒\Rightarrow 2r + r θ\theta = 20

⇒\Rightarrow θ=20−2rr\theta = {{20 - 2r} \over r} .......(1)

A = Area = θ2π×πr2{\theta \over {2\pi }} \times \pi {r^2} = 12r2θ{1 \over 2}{r^2}\theta = 12r2(20−2rr){1 \over 2}{r^2}\left( {{{20 - 2r} \over r}} \right)

⇒\Rightarrow A = 10r – r2

For A to be maximum

dAdr=0{{dA} \over {dr}} = 0

⇒\Rightarrow 10 – 2r = 0

⇒\Rightarrow r = 5

{{{d^2}A} \over {d{r^2}}} = - 2 < 0

∴\therefore For r = 5, A is maximum From (1)

θ=20−2(5)r{\theta = {{20 - 2\left( 5 \right)} \over r}} = 2

A = 22π×π(5)2{2 \over {2\pi }} \times \pi {\left( 5 \right)^2} = 25 sq. m

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