1. **State the problem:** We have a right triangle with an angle of $83^\circ$, the side opposite this angle is $4'11$ (4 feet 11 inches), the adjacent side is $26'5$ (26 feet 5 inches), and we need to find the hypotenuse (?). 2. **Convert all measurements to a consistent unit (inches):** - $4'11 = 4 \times 12 + 11 = 48 + 11 = 59$ inches - $26'5 = 26 \times 12 + 5 = 312 + 5 = 317$ inches 3. **Use the Law of Cosines:** Since we have two sides and the included angle, the Law of Cosines is appropriate: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$ where $c$ is the hypotenuse, $a = 59$, $b = 317$, and $C = 83^\circ$. 4. **Calculate:** $$c^2 = 59^2 + 317^2 - 2 \times 59 \times 317 \times \cos(83^\circ)$$ Calculate each term: - $59^2 = 3481$ - $317^2 = 100489$ - $\cos(83^\circ) \approx 0.12187$ So, $$c^2 = 3481 + 100489 - 2 \times 59 \times 317 \times 0.12187$$ Calculate the product: $$2 \times 59 \times 317 = 37306$$ Then, $$37306 \times 0.12187 \approx 4546.5$$ Therefore, $$c^2 = 3481 + 100489 - 4546.5 = 103970.5$$ 5. **Find $c$ by taking the square root:** $$c = \sqrt{103970.5} \approx 322.4 \text{ inches}$$ 6. **Convert back to feet and inches:** - Feet: $\lfloor 322.4 / 12 \rfloor = 26$ feet - Inches: $322.4 - 26 \times 12 = 322.4 - 312 = 10.4$ inches **Final answer:** The hypotenuse is approximately $26'10.4$ (26 feet 10.4 inches).