Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.

Step-by-step explanation:

Let the rectangle have (x,y,z) as vertex in positive octant.  The rectangular box has to be necessarily symmetrical about all the three axes.

Then the sides of the box would be

Volume =

Maximize volume subject to

i.e.

Use Lagrangian multipliers , we have

at the maximum

Dividing we get

Similarly

Thus we get

Hence dimensions are

(2x,2y,2z)

So dimensions are

may i see the picture? it would a lot, xox