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  • 35. An open box with square base is to be made of a given quantity of card board of area c^2 . Show that the  maximum volume of the box is 6root3 cubic units 

35. An open box with square base is to be made of a given quantity of card board of area c^2 . Show that the  maximum volume of the box is 6root3 cubic units 

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Let the side of the square base be x and the height be h
As we know, Surface area = area of base + area of 4 sides

Given: x^2+4 x h=c^2

Therefore, h=\frac{c^2-x^2}{4 x}

Now, the volume of the box will be, V=x^2 h=x^2 \times \frac{c^2-x^2}{4 x}=\frac{c^2 x-x^3}{4}

Upon differentiating V with respect to x we get,

\frac{d V}{d x}=\frac{c^2-3 x^2}{4}

For calculating maximum volume, set \frac{d V}{d x}=0

c^2-3 x^2=0 \quad \Rightarrow \quad x=\frac{c}{\sqrt{3}}

Upon substituting x=\frac{c}{\sqrt{3}} into the volume

V=\frac{c^2 \times \frac{c}{\sqrt{3}}-\left(\frac{c}{\sqrt{3}}\right)^3}{4}=\frac{c^3 / \sqrt{3}-c^3 / 3 \sqrt{3}}{4}=\frac{\left(2 c^3\right) /(3 \sqrt{3})}{4}=\frac{c^3}{6 \sqrt{3}}

Therefore, the maximum volume will be, \frac{c^3}{6 \sqrt{3}}

 

Posted by

Divya Sharma

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