8. In \( \triangle \mathrm{STU}, u=8.9 \) inches, \( m \angle \mathrm{U}=43^{\circ} \) and \( m \angle \mathrm{~S}=99^{\circ} \). Find the length of \( s \), to the nearest 10 th of an inch.

To find the length of side \( s \) in triangle \( \triangle STU \), we can use the Law of Sines, which states: \[ \frac{s}{\sin U} = \frac{u}{\sin S} \] First, we need to find angle \( T \): \[ m \angle T = 180^\circ - m \angle S - m \angle U = 180^\circ - 99^\circ - 43^\circ = 38^\circ \] Now, applying the Law of Sines: \[ \frac{s}{\sin 43^\circ} = \frac{8.9}{\sin 99^\circ} \] Since \( \sin 99^\circ \) is approximately equal to \( \sin 81^\circ \), we can compute: \[ \frac{s}{\sin 43^\circ} = \frac{8.9}{\sin 81^\circ} \] Calculating \( \sin 43^\circ \approx 0.681998 \) and \( \sin 81^\circ \approx 0.980785 \): \[ \frac{s}{0.681998} = \frac{8.9}{0.980785} \] Solving for \( s \): \[ s = \frac{8.9 \times 0.681998}{0.980785} \approx 6.0705 \] To the nearest tenth of an inch, the length of \( s \) is \( 6.1 \) inches.