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.

We are given triangle RST with vertices R, S, and T. The side labels are defined so that side r is opposite angle R, side s is opposite angle S, and side t is opposite angle T. The given values are:  • t = 53 inches (side RS is opposite vertex T)  • r = 81 inches (side ST is opposite vertex R)  • angle S = 51° (vertex S, where sides RS and ST meet) The side we need to find is s (side RT), which is opposite angle S. Because angle S is the included angle between sides t and r, we can use the Law of Cosines:   s² = r² + t² − 2·r·t·cos(S) Step 1. Substitute the known values:   s² = 81² + 53² − 2·81·53·cos 51° Step 2. Calculate the squares:   81² = 6561   53² = 2809   6561 + 2809 = 9370 Step 3. Calculate the product term:   2·81·53 = 8586 Step 4. Write the equation:   s² = 9370 − 8586·cos 51° Step 5. Using a calculator, find cos 51°. (cos 51° ≈ 0.62932)   8586 × 0.62932 ≈ 5400 Thus,   s² ≈ 9370 − 5400 = 3970 Step 6. Find s by taking the square root:   s ≈ √3970 ≈ 63 inches Rounding to the nearest inch, the length of side s is 63 inches.