1. **Problem statement:** Diane has a triangular backyard with two fence posts. The distances from the deck to the fence posts are 90 m and 82 m, and the angle between these two lines from the deck is 88°. We need to find the distance between the two fence posts (side $b$). 2. **Formula used:** To find the length of a side in a triangle when two sides and the included angle are known, we use the Law of Cosines: $$b^2 = a^2 + c^2 - 2ac \cos(B)$$ where $a=90$, $c=82$, and $B=88^\circ$. 3. **Calculate $b^2$:** $$b^2 = 90^2 + 82^2 - 2 \times 90 \times 82 \times \cos(88^\circ)$$ 4. **Evaluate each term:** $$90^2 = 8100$$ $$82^2 = 6724$$ $$2 \times 90 \times 82 = 14760$$ $$\cos(88^\circ) \approx 0.0349$$ 5. **Substitute values:** $$b^2 = 8100 + 6724 - 14760 \times 0.0349$$ $$b^2 = 14824 - 515.724$$ $$b^2 = 14308.276$$ 6. **Find $b$ by taking the square root:** $$b = \sqrt{14308.276} \approx 119.6$$ 7. **Answer:** The distance between the fence posts is approximately **119.6 metres**. This completes the solution for the first problem.