1. **State the problem:** We have triangle $\triangle STU$ with side $s = 250$ inches opposite angle $S$, angle $\angle U = 17^\circ$, and angle $\angle S = 52^\circ$. We need to find the length of side $t$ opposite angle $T$. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. $$\angle T = 180^\circ - \angle S - \angle U = 180^\circ - 52^\circ - 17^\circ = 111^\circ$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{s}{\sin(\angle S)} = \frac{t}{\sin(\angle T)}$$ 4. **Plug in known values:** $$\frac{250}{\sin(52^\circ)} = \frac{t}{\sin(111^\circ)}$$ 5. **Solve for $t$:** Multiply both sides by $\sin(111^\circ)$: $$t = \frac{250}{\sin(52^\circ)} \times \sin(111^\circ)$$ 6. **Calculate sine values:** $$\sin(52^\circ) \approx 0.7880$$ $$\sin(111^\circ) \approx 0.9336$$ 7. **Substitute and simplify:** $$t = \frac{250}{0.7880} \times 0.9336$$ 8. **Intermediate step with cancellation:** $$t = 250 \times \frac{0.9336}{\cancel{0.7880}} \times \cancel{\frac{1}{0.7880}}$$ 9. **Calculate final value:** $$t \approx 250 \times 1.1849 = 296.2$$ 10. **Answer:** The length of side $t$ is approximately **296.2 inches** to the nearest tenth.