We know that the square base of the new box is 4 sq. in. This means that each side has to equal 2 in., because A = l*w and 4 = 2x2; we also know that since it is a square, both sides have to be equal, so it must be a 2x2 square and not a 4x1 rectangle.
That being said, the first box had dimensions of 4x4x4 - and this is confirmed by the volume being 64 sq. in. and the height being told to us as 4 in. as well as the square base having to have equal sides.
Because of this, since the question states "similar," all of the dimensions have to be reduced by the same factor. Since the square base was reduced by 1/2, we can assume the height is also reduced by 1/2, thus giving us a height of the new box as 2 in.
So, the volume of the new box is 2x2x2, which equals 8 sq. in.
