1. **Problem 1: Calculate the round trip distance Michelle traveled on her motorcycle.** Given: Triangle with sides 45 miles and 32 miles, and included angle 70° between them. 2. **Formula used:** To find the third side of a triangle when two sides and the included angle are known, use the Law of Cosines: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $a=45$, $b=32$, and $C=70^\circ$. 3. **Calculate the third side $c$: ** $$c^2 = 45^2 + 32^2 - 2 \times 45 \times 32 \times \cos(70^\circ)$$ Calculate each term: $$45^2 = 2025$$ $$32^2 = 1024$$ $$2 \times 45 \times 32 = 2880$$ $$\cos(70^\circ) \approx 0.3420$$ So, $$c^2 = 2025 + 1024 - 2880 \times 0.3420 = 3049 - 984.96 = 2064.04$$ 4. **Find $c$: ** $$c = \sqrt{2064.04} \approx 45.44$$ miles 5. **Calculate total round trip distance:** Michelle traveled from Town A to Town B (45 miles), Town B to Town C (32 miles), and Town C back to Town A (the side $c$ we just found). Total distance = $45 + 32 + 45.44 = 122.44$ miles 6. **Final answer:** Michelle's round trip distance is approximately **122.4 miles** to the nearest tenth. 2. **Problem 2: Find the distance across the base of the roof.** Given: Triangle with sides 12 feet and 28 feet, and included angle 110°. 3. **Use Law of Cosines to find the base $b$: ** $$b^2 = 12^2 + 28^2 - 2 \times 12 \times 28 \times \cos(110^\circ)$$ Calculate each term: $$12^2 = 144$$ $$28^2 = 784$$ $$2 \times 12 \times 28 = 672$$ $$\cos(110^\circ) \approx -0.3420$$ So, $$b^2 = 144 + 784 - 672 \times (-0.3420) = 928 + 229.82 = 1157.82$$ 4. **Find $b$: ** $$b = \sqrt{1157.82} \approx 34.03$$ feet 5. **Final answer:** The distance across the base of the roof is approximately **34.0 feet** to the nearest tenth.