Answer
The distance between the centers of atoms A and C is 11.6
Work Step by Step
We can find angle $C$:
$\frac{a}{sin~A} = \frac{c}{sin~C}$
$sin~C = \frac{c~sin~A}{a}$
$C = arcsin(\frac{c~sin~A}{a})$
$C = arcsin(\frac{(3.0+4.5)~sin~(18^{\circ})}{2.0+3.0})$
$C = arcsin(\frac{(7.5)~sin~(18^{\circ})}{5.0})$
$C = 27.6^{\circ}$
We can find angle $B$:
$A+B+C = 180^{\circ}$
$B = 180^{\circ}-A-C$
$B = 180^{\circ}-18^{\circ}-27.6^{\circ}$
$B = 134.4^{\circ}$
We can find the length of side $b$:
$\frac{b}{sin~B} = \frac{a}{sin~A}$
$b = \frac{a~sin~B}{sin~A}$
$b = \frac{(5.0)~sin~(134.4^{\circ})}{sin~(18^{\circ})}$
$b = 11.6$
The distance between the centers of atoms A and C is 11.6
