Question

Of all open tanks in the form of a rectangular parallelopiped with a square.base having a fixed volume, the one that has the least inner surface area has the property that its

• A. depth in half of the base side
• B. depth is twice the base side
• C. depth is equal to the base side
• D. depth is one-third of the base side

Answer 1

Answer: (C)

depth is equal to the base side

Answer 2

Let length, breadth and height of rectangular parallelepiped are $I, I$ and $h$ respectively.
Volume $(V) = l^2h \, \, \, \, \, \, \, \, \, \, \,$ ...(i)
and surface area, $(S) = 2(l^2+ hl + lh) S = 2l^2 + 4hl \, \, \, \, \, \, \, \,$.....(ii)
differentiating (i) w.r.t. h, we.get
$\frac{dV}{dh} =2lh \frac{dl}{dh}+l^{2}$
$\Rightarrow 2lh \frac{dl}{dh} + l^{2} =0$
$\Rightarrow \frac{dl}{dh} =- \frac{l}{2h}$