Let l, w, and h be the length, width, and height of the rectangular prism, respectively. We know that the total surface area is 56, so we can use the formula for the surface area of a rectangular prism to get the following equation:
2lw+2wh+2lh=56
We also know that the sum of all the edges is 64, so we can use the formula for the sum of the edges of a rectangular prism to get the following equation:
2l+2w+2h=64
We can solve the first equation for l to get:
l=2w56−2wh−2lh
We can then substitute this into the second equation to get:
2w56−2wh−2lh+2w+2h=64
We can then simplify this equation to get:
2wh+2lh−56=0
We can then factor this equation to get:
(w−4)(2h−14)=0
We can then solve for w and h to get:
w=4 h=214=7
We can then use these values to find the length of the diagonal by using the Pythagorean theorem:
d2=l2+w2+h2=42+72+72=108
d=sqrt(108)=6*sqrt(3)
Therefore, the length of the diagonal joining one corner of the prism to the opposite corner is 6*sqrt(3).
