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
Here's the solution for question 9. a) Step 1: Apply the Cosine Rule to find . Given OM = 6 cm, ON = 6 cm, and MN = 8 cm. The cosine rule states c^2 = a^2 + b^2 - 2ab C. In OMN, we have: MN^2 = OM^2 + ON^2 - 2(OM)(ON) Substitute the given values: 8^2 = 6^2 + 6^2 - 2(6)(6) 64 = 36 + 36 - 72 64 = 72 - 72 Rearrange to solve for : 72 = 72 - 64 72 = 8 = (8)/(72) = (1)/(9) Calculate : = ((1)/(9)) ≈ 83.6206^ Correct to the nearest whole number: = 84^ b) Step 2: Calculate the area of the shaded region. The shaded region is the area of the sector OMN minus the area of the triangle OMN. Radius of the sector r = 6 cm. Angle ≈ 83.6206^. Use = (22)/(7). First, calculate the area of the sector OMN: Area of sector = ()/(360^) × r^2 Area of sector = (83.6206)/(360) × (22)/(7) × (6)^2 Area of sector = (83.6206)/(360) × (22)/(7) × 36 Area of sector ≈ 26.279 cm^2 Next, calculate the area of the triangle OMN: Area of triangle = (1)/(2) OM · ON Area of triangle = (1)/(2) (6)(6) (83.6206^) Area of triangle = 18 (83.6206^) Area of triangle ≈ 18 × 0.9937 Area of triangle ≈ 17.8866 cm^2 Finally, calculate the area of the shaded region: Area of shaded region = Area of sector - Area of triangle Area of shaded region = 26.279 - 17.8866 Area of shaded region = 8.3924 cm^2 Correct to two significant figures: Area of shaded region = 8.4 cm^2 3 done, 2 left today. You're making progress.