What is the smallest volume?

Question

of golden chest that encloses emerald egg that
encloses amber needle of length 3 inches?

Answer 1

You need more information. We can assume that an egg that ecloses a needle (I'm assuming a shape like a sewing needle) that's 3 inches long must itself be at least 3 inches. But in theory it seems that the egg could be of any other dimesions, which means the box could be too.

Answer 2

This is a complicated question involving many unknowns, such as the state of the Asker's mind, but seeing that the chest is golden, the ratios has to be that of a golden cuboid, so therefore, the volume is 1 times some factor yet to be determined.

Answer 3

let's assume the thickness of the egg and the chest gotta be ~0.02 inch (about half a millimeter, sounds about right).

now let's assume that for an egg is an ellipsoid, and not too narrow an ellipsoid - let's define an "egg" as an ellipsoid whose radii are 2x1x1

then, volume of the egg =
4/3 PI (1.52 * 0.76 * 0.76) = 3.678 inch cube

now the golden chest's dimensions are
3.08 x 1.58 x 1.58, volume = 7.689 inch cube

Answer 4

for us to have the smallest volume, the egg can be assumed as a sphere with a radius of 1.5inches (3 inches for the diameter)

Volume of a sphere = 4/3Pir^3
= 4/3 * 22/7 * (1.5)^3
= 14.1428 cubic inches