1. **State the problem:** We are given triangle $\triangle STU$ with side $t = 2.7$ inches, angle $\angle T = 81^\circ$, and angle $\angle U = 72^\circ$. We need to find the area of the triangle to the nearest tenth of a square inch. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. So, $$\angle S = 180^\circ - 81^\circ - 72^\circ = 27^\circ.$$ 3. **Use the Law of Sines to find other sides:** The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$$ Here, side $t$ is opposite angle $T$, so $t$ corresponds to side $TU$ opposite $\angle T$. Let side $s$ be opposite $\angle S$ and side $u$ opposite $\angle U$. We know $t = 2.7$ inches opposite $81^\circ$. Calculate side $s$ opposite $27^\circ$: $$\frac{s}{\sin 27^\circ} = \frac{2.7}{\sin 81^\circ} \implies s = \frac{2.7 \sin 27^\circ}{\sin 81^\circ}.$$ Calculate side $u$ opposite $72^\circ$: $$\frac{u}{\sin 72^\circ} = \frac{2.7}{\sin 81^\circ} \implies u = \frac{2.7 \sin 72^\circ}{\sin 81^\circ}.$$ 4. **Calculate the area using formula:** Area of triangle = $\frac{1}{2} \times (side_1) \times (side_2) \times \sin(\text{included angle})$. We can use sides $s$ and $u$ with included angle $\angle T = 81^\circ$: $$\text{Area} = \frac{1}{2} s u \sin 81^\circ.$$ 5. **Substitute values and compute:** Calculate $s$: $$s = \frac{2.7 \times \sin 27^\circ}{\sin 81^\circ} = \frac{2.7 \times 0.45399}{0.98769} = \frac{1.22577}{0.98769} = 1.241.$$ Calculate $u$: $$u = \frac{2.7 \times \sin 72^\circ}{\sin 81^\circ} = \frac{2.7 \times 0.95106}{0.98769} = \frac{2.56786}{0.98769} = 2.599.$$ Calculate area: $$\text{Area} = \frac{1}{2} \times 1.241 \times 2.599 \times \sin 81^\circ = 0.5 \times 1.241 \times 2.599 \times 0.98769 = 1.592.$$ 6. **Round to nearest tenth:** Area $\approx 1.6$ square inches. **Final answer:** The area of $\triangle STU$ is approximately $\boxed{1.6}$ square inches.