This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Calculate angle PRQ. We use the Sine Rule: (PQ)/(( PRQ)) = (PR)/(( PQR)) Substitute the given values: PQ = 41.6 m, PR = 70 m, PQR = 105^. (41.6)/(( PRQ)) = (70)/( 105^) Rearrange to solve for ( PRQ): ( PRQ) = (41.6 × 105^)/(70) Calculate 105^ ≈ 0.9659258: ( PRQ) = (41.6 × 0.9659258)/(70) ( PRQ) = (40.20161328)/(70) ( PRQ) ≈ 0.57430876 Now, find PRQ: PRQ = (0.57430876) PRQ ≈ 35.05^ Part (i): Calculate angle QPR. The sum of angles in a triangle is 180^. QPR + PQR + PRQ = 180^ QPR = 180^ - PQR - PRQ Substitute the values: QPR = 180^ - 105^ - 35.05^ QPR = 39.95^ Rounding to one decimal place: QPR ≈ 40.0^ Part (ii): Calculate the area of triangle PQR. The area of a triangle can be calculated using the formula: Area = (1)/(2)ab C Using sides PQ, PR and the included angle QPR: Area = (1)/(2) × PQ × PR × ( QPR) Substitute the values: PQ = 41.6 m, PR = 70 m, QPR = 39.95^. Area = (1)/(2) × 41.6 m × 70 m × (39.95^) Calculate (39.95^) ≈ 0.64207: Area = (1)/(2) × 41.6 × 70 × 0.64207 Area = 20.8 × 70 × 0.64207 Area = 1456 × 0.64207 Area ≈ 934.78 m^2 Rounding to one decimal place: Area ≈ 934.8 m^2 Part (iii): Calculate the shortest distance from Q to PR. The shortest distance from a vertex to the opposite side is the altitude (height). Let h be the shortest distance from Q to PR. The base is PR. The area of a triangle can also be calculated as: Area = (1)/(2) × base × height Area = (1)/(2) × PR × h We know the Area from Part (ii) and PR = 70 m. 934.78 m^2 = (1)/(2) × 70 m × h 934.78 = 35 × h Rearrange to solve for h: h = (934.78)/(35) h ≈ 26.708 m Rounding to one decimal place: h ≈ 26.7 m That's 2 down. 3 left today — send the next one.