Surds - Applications of Surds

The diagram below shows a pencil case which is a cuboid. \(AB=\left( 5\sqrt{7}+2 \right)\) cm, \(BC=6\)cm and \(CF=5\)cm.

1. Find, without using a calculator, the exact value of \({{(AC)}^{2}}\)in the form \(\left( a+b\sqrt{7} \right)\mathrm{ c}{{\mathrm{m}}^{\mathrm{2}}}\).
2. Hence find the length of the longest pencil that can be placed inside the pencil case, giving your answer correct to one decimal place.