CEMC Fryer, 2008

(a) Figure 1 shows a net that can be folded to create a rectangular box, with dimensions $4$, $4$, and $7$ as labeled on the net. Determine the volume and the surface area of the box.

(b) In Figure 2, the rectangular box has dimensions $2$ by $2$ by $6$. From point $A$, an ant walked to point $B$ crossing all four of the side faces. The shortest path along which the ant could walk may be found by unfolding the box, as in Figure 3, and drawing a straight line from $A$ to $B$. Determine the length of $AB$ in Figure 3.

(c) In Figure 4, the rectangular block has dimensions $3$ by $4$ by $5$. A caterpillar is at corner $A$. Determine, with justification, the shortest possible distance from $A$ to $G$ along the surface of the block.