2. In \( \triangle \mathrm{RST}, t=53 \) inches, \( r=81 \) inches and \( \angle \mathrm{S}=51^{\circ} \). Find the length of \( s \), to the nearest inch.
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To find the length of side \( s \) in triangle \( \triangle RST \) given the lengths of sides \( r \) and \( t \) and the angle \( \angle S \), we can use the Law of Cosines. The formula is: \[ s^2 = r^2 + t^2 - 2rt \cos(S) \] Substituting the known values: \[ s^2 = 81^2 + 53^2 - 2 \times 81 \times 53 \times \cos(51^\circ) \] Calculating each part: \[ 81^2 = 6561, \quad 53^2 = 2809, \quad \cos(51^\circ) \approx 0.6293 \] Thus, \[ s^2 = 6561 + 2809 - 2 \times 81 \times 53 \times 0.6293 \] \[ s^2 = 6561 + 2809 - 2 \times 81 \times 53 \times 0.6293 \approx 6561 + 2809 - 5220.14 \] \[ s^2 \approx 6561 + 2809 - 5220.14 \approx 4149.86 \] \[ s \approx \sqrt{4149.86} \approx 64.31 \] Rounding to the nearest inch, we find: \[ s \approx 64 \text{ inches} \] So the length of side \( s \) is approximately 64 inches.
