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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 :
Option: 1 \begin{aligned} &\frac{a+b-\sqrt{a^{2}+b^{2}-a b}}{6}\\ \end{aligned}
Option: 2 \frac{a+b-\sqrt{a^{2}+b^{2}-a b}}{12}\\
Option: 3 \frac{a+b+\sqrt{a^{2}+b^{2}-a b}}{6}\\
Option: 4 \frac{a+b-\sqrt{a^{2}+b^{2}+a b}}{6}

Answers (1)

best_answer

The volume of box  = x\left ( a-2x \right )\left ( b-2x \right )\\

V(x)= 4x^{3}-2x^{2}\left ( a+b \right )+abx= 0\\

V'(x)= 12x^{2}-4x\left ( a+b \right )+ab= 0\\

\Rightarrow x= \frac{4\left ( a+b \right )\pm 4\sqrt{\left ( a+b \right )^{2}-3ab}}{24}\\

= \frac{\left ( a+b \right )\pm \sqrt{a^{2}+b^{2}-ab}}{6}\\

Point of maxima will be x= \frac{\left ( a+b \right )\sqrt{a^{2}+b^{2}-ab}}{6}\\

Posted by

Kuldeep Maurya

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