Steven,

The longest rod that will just fit inside a rectangular box, if placed diagonally top to bottom, is 17 inches.

Using Pythagoras' theorem in its three-dimensional version

D² = L² + W² + H² (Diagonal, Length, Width, Height)

write the diagonal length of the rod [as a number] in terms of length, width, and height.

The box is 1 inch shorter and 3 inches longer than it is wide.

That's poorly phrased; note that "short" here is opposed to "tall" not to "long". So write height and length in terms of width.

Substitute this into your first equation to get the right hand side in terms of width alone. Solve for W.

How much must you cut off the rod so that it will lie flat in the bottom of the container? What are the dimensions of the box?

Find L. Find the diagonal of the bottom. Subtract. Find H.

Go get a snack.

Good Hunting!
RD