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.
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Here are the calculations for the problem: Part (a)(i): Calculate PR Step 1: Determine the interior angle PQR. The bearing of Q from P is 050^. This means the angle from the North line at P, clockwise to PQ, is 50^. Since the North lines at P and Q are parallel, the alternate interior angle between the North line at Q (pointing North) and QP is 50^. The bearing of R from Q is 130^. This is the angle from the North line at Q, clockwise to QR. The angle from the North line at Q, clockwise to the South line at Q is 180^. Therefore, the angle between the South line at Q and QR (measured anti-clockwise from South) is 180^ - 130^ = 50^. The angle PQR is the sum of the angle between QP and the North line at Q (west side) and the angle between QR and the South line at Q (west side). PQR = 50^ + 50^ = 100^ Step 2: Calculate the distance PR using the Cosine Rule. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. Here, PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)( PQR). PR^2 = 8^2 + 13^2 - 2(8)(13)(100^) PR^2 = 64 + 169 - 208(-0.1736) PR^2 = 233 + 36.11878 PR^2 = 269.11878 PR = sqrt(269.11878) ≈ 16.40489 m Rounding to three significant figures: PR ≈ 16.4 m Part (a)(ii): Calculate the bearing of R from P Step 3: Calculate QPR using the Sine Rule. The Sine Rule states (a)/( A) = (b)/( B). Here, (QR)/(( QPR)) = (PR)/(( PQR)). (13)/(( QPR)) = (16.40489)/((100^)) ( QPR) = (13 (100^))/(16.40489) ( QPR) = (13 × 0.98480775)/(16.40489) ≈ 0.77980 QPR = (0.77980) ≈ 51.25^ Step 4: Calculate the bearing of R from P. The bearing of Q from P is 050^. The bearing of R from P is the bearing of Q from P plus QPR. Bearing of R from P = 50^ + 51.25^ = 101.25^ Rounding to three significant figures: Bearing of R from P ≈ 101^ Part (b): Calculate the shortest distance between point Q and line PR, hence the area of triangle PQR. Step 5: Calculate the shortest distance from Q to PR. The shortest distance is the altitude (height) from Q to the base PR. Let this distance be h. In the right-angled triangle formed by Q, P, and the foot of the perpendicular from Q to PR, we have: h = PQ ( QPR) h = 8 (51.25^) h = 8 × 0.7800 ≈ 6.240 m Rounding to three significant figures: Shortest distance ≈ 6.24 m Step 6: Calculate the area of triangle PQR. The area of a triangle can be calculated using the formula: Area = (1)/(2)ab C. Area of PQR = (1)/(2) (PQ)(QR)( PQR) Area of PQR = (1)/(2) (8)(13)(100^) Area of PQR = 52 × 0.98480775 Area of PQR ≈ 51.20999 m^2 Rounding to three significant figures: Area of PQR ≈ 51.2 m^2 Drop the next question! 📸