Question

If x,y and z are the dimensions of cuboid ,then find among x,y and z which will increase the volume of cuboid when after adding one unit

Answer 1

First, assume z is some constant. Let s = sqrt(xy) and a = sqrt(x/y). Then x = sa and y = s/a.Holding s constant, then no matter what a is, the volume is still xyz = s^2 * z.We want to show that the surface area is minimized when a = 1, i.e. x = y.

Surface Area /2 = xy + xz + yz = s^2 + z(x+y) = s^2 +zs(a + 1/a). Since z and s are constant, we need to minimize (a + 1/a). The derivative is (1-a^-2), which is zero at a=1.The second derivative, 2*a^-3 = 2 (at a =1), which is positive, so this is a minimum.

Thus elementary calculus shows that whatever z is, to minimize surface area, x must equal y.By symmetry, whatever y is, z must equal x; and whatever x is, z must equal y.These conditions can be satisfied only if all three are equal.