If we want to make a cuboid with a square base, how can we save the material most to make the cuboid with a capacity of 108 cube meters (the cuboid has no cover).

In order to solve this question, we have to find out the volume and the total surface of the cuboid. Let x be the side of the base, and y be the length of the cuboid. In this case, we know the volume equals to 108 which equals to (x^2)(h). So we have h=108/x^2. In addition, we know the surface of the cuboid is x^2+4xh where h =108/x^2. So, we can get the equation that S=x^2+432/x. If we derivative it, we can get 2x-432x^-2. If we can find the critical point, we can find the x and h that save the most materials; therefore, making 2x-432x^-2=0, we can get x equals to 6 and h equals to 3.

In conclusion, when the sides of the base equal to 6m and the height equals to 3m would save the most materials to make a cuboid.