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QJEEM21Ma18a7: JEE Main 2021 Mathematics - Class 12 JEEM21Ma18a7 Solution: A box open from top is made from a rectangular sheet of…

A box open from top is made from a rectangular sheet of dimension a ×\times b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

Multiple Choice Options

  • A a+b−a2+b2−ab12{{a + b - \sqrt {{a^2} + {b^2} - ab} } \over {12}}
  • B a+b−a2+b2+ab6{{a + b - \sqrt {{a^2} + {b^2} + ab} } \over 6}
  • C a+b−a2+b2−ab6{{a + b - \sqrt {{a^2} + {b^2} - ab} } \over 6}
  • D a+b+a2+b2+ab6{{a + b + \sqrt {{a^2} + {b^2} + ab} } \over 6}

Step-by-Step Verified Solution

JEE Main 2021 (Online) 27th August Evening Shift Mathematics - Application of Derivatives Question 89 English Explanation 1

V = l . b . h = (a −- 2x)(b −- 2x) x

⇒\Rightarrow V(x) = (2x −- a)(2x −- b) x

⇒\Rightarrow V(x) = 4x3 −- 2(a + b)x2 + abx

⇒ddxv(x)=12x2−4(a+b)x+ab\Rightarrow {d \over {dx}}v(x) = 12{x^2} - 4(a + b)x + ab

{d \over {dx}}(v(x)) = 0 \Rightarrow 12{x^2} - 4(a + b)x + ab = 0 < _\beta ^\alpha

⇒x=4(a+b)±16(a+b)2−48ab2(12)\Rightarrow x = {{4(a + b) \pm \sqrt {16{{(a + b)}^2} - 48ab} } \over {2(12)}}

=(a+b)±a2+b2−ab6= {{(a + b) \pm \sqrt {{a^2} + {b^2} - ab} } \over 6}

Let x=α=(a+b)+a2+b2−ab6x = \alpha = {{(a + b) + \sqrt {{a^2} + {b^2} - ab} } \over 6}

β=(a+b)−a2+b2−ab6\beta = {{(a + b) - \sqrt {{a^2} + {b^2} - ab} } \over 6}

Now, 12(x−α)(x−β)=012(x - \alpha )(x - \beta ) = 0

JEE Main 2021 (Online) 27th August Evening Shift Mathematics - Application of Derivatives Question 89 English Explanation 2

∴\therefore x = β\beta =a+b−a2+b2−abb= {{a + b - \sqrt {{a^2} + {b^2} - ab} } \over b}

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