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
a) i) Calculate KN Step 1: Use the Cosine Rule to find the length of side KN. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. KN^2 = KM^2 + MN^2 - 2(KM)(MN) ( KMN) KN^2 = 8^2 + 10^2 - 2(8)(10) (92^) KN^2 = 64 + 100 - 160 (92^) Step 2: Calculate the value. KN^2 = 164 - 160(-0.034899) KN^2 = 164 + 5.58384 KN^2 = 169.58384 Step 3: Take the square root to find KN. KN = sqrt(169.58384) KN ≈ 13.0224 km Rounding to three significant figures: KN = 13.0 km ii) Calculate the area of triangle KMN Step 1: Use the formula for the area of a triangle given two sides and the included angle: Area = (1)/(2)ab C. Area = (1)/(2) (KM)(MN) ( KMN) Area = (1)/(2) (8 km)(10 km) (92^) Step 2: Calculate the area. Area = 40 (92^) Area = 40 × 0.99939 Area ≈ 39.9756 km^2 Rounding to three significant figures: Area = 40.0 km^2 iii) Calculate the shortest distance from M to KN Step 1: The shortest distance from a point to a line is the perpendicular height. Use the area formula: Area = (1)/(2) × base × height. Area = (1)/(2) × KN × h 39.9756 km^2 = (1)/(2) × 13.0224 km × h Step 2: Solve for h. h = (2 × 39.9756)/(13.0224) h ≈ 6.1394 km Rounding to three significant figures: h = 6.14 km b) Solve the equation 2 = -3 for 0^ 180^ Step 1: Isolate . 2 = -3 = -(3)/(2) = -1.5 Step 2: Find the reference angle, , using the absolute value of . = (1.5) ≈ 56.3099^ Step 3: Determine the quadrant for . Since is negative and the range is 0^ 180^, must be in the second quadrant. Step 4: Calculate in the second quadrant. = 180^ - = 180^ - 56.3099^ = 123.6901^ Rounding to one decimal place: = 123.7^ Drop the next question!