Deer Distance A22Dad

1. **State the problem:** A deer walks $180$ ft east, then turns $112^\circ$ clockwise and walks $105$ ft. We need to find how far the deer is from its starting point. 2. **Understand the situation:** The deer first walks straight east, then changes direction by turning $112^\circ$ clockwise and walks $105$ ft. We want the straight-line distance from start to end. 3. **Use the Law of Cosines:** The path forms a triangle with sides $180$ ft and $105$ ft, and the angle between them is $112^\circ$. The Law of Cosines formula is: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $a=180$, $b=105$, and $C=112^\circ$. 4. **Calculate:** $$c^2 = 180^2 + 105^2 - 2 \times 180 \times 105 \times \cos(112^\circ)$$ Calculate each term: $$180^2 = 32400$$ $$105^2 = 11025$$ Calculate $\cos(112^\circ)$ (approx): $$\cos(112^\circ) \approx -0.3746$$ Substitute: $$c^2 = 32400 + 11025 - 2 \times 180 \times 105 \times (-0.3746)$$ Calculate the product: $$2 \times 180 \times 105 = 37800$$ So: $$c^2 = 32400 + 11025 + 37800 \times 0.3746$$ $$c^2 = 32400 + 11025 + 14156.28$$ $$c^2 = 57581.28$$ 5. **Find $c$:** $$c = \sqrt{57581.28} \approx 239.9$$ 6. **Answer:** The deer is approximately $239.9$ ft from its starting point.