Answer
The distance across the river is 293.4 meters
Work Step by Step
Let $A$ be Mark's position. Then angle $A = 180^{\circ} - 115.45^{\circ}$ which is $64.55^{\circ}$.
Let $C$ be Lisa's position. Then angle $C = 45.47^{\circ}$.
Let the tree be located at the position of angle $B$. We can find angle $B$:
$A+B+C = 180^{\circ}$
$B = 180^{\circ}-A-C$
$B = 180^{\circ}-64.55^{\circ}-45.47^{\circ}$
$B = 69.98^{\circ}$
We can find the length of side $a$ which is the distance from Lisa to the tree:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$a = \frac{b~sin~A}{sin~B}$
$a = \frac{(428.3~m)~sin~(64.55^{\circ})}{sin~(69.98^{\circ})}$
$a = 411.6~mi$
Let angle $\theta$ be the angle between a horizontal line across the river and the line from Lisa to the tree. We can find $\theta$:
$\theta = 90^{\circ} - 45.47^{\circ} = 44.53^{\circ}$
We can find the distance $d$ across the river:
$\frac{d}{a} = cos~\theta$
$d = a~cos~\theta$
$d = (411.6~m)~cos~(44.53^{\circ})$
$d = 293.4~m$
The distance across the river is 293.4 meters
