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OR An open box with a square base is to ...

OR An open box with a square base is to be made out of a given quantity of cardboard of area`\ c^2` square units. Show that the maximum volume of the box is `(c^3)/(6\ sqrt(3))` cubic units.

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Let each side of the base be a and height be h. Then,
`c^(2) = (a^(2) + 4ah) rArr h = ((c^(2) -a^(2)))/(4a)`
So, `V = a^(2) xx h rArr V = ((c^(2) a - a^(3)))/(4)`
`:. (dV)/(da) = ((C^(2) - 3a^(2)))/(4) and (d^(2)V)/(da^(2)) = - (3)/(2) a lt 0`
Thus, V is maximum when `a^(2) = (c^(2))/(3) and h = (c)/(2 sqrt3)`
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