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 to question 11: a) Calculate angle ABC. Step 1: Apply the Sine Rule. The Sine Rule states that for a triangle with sides a, b, c and opposite angles A, B, C: (a)/( A) = (b)/( B) = (c)/( C) In ABC, we have: Side a = BC = 35.3 cm Angle A = CAB = 51.6^ Side b = AC = 29.5 cm Angle B = ABC (which we need to find) Using the Sine Rule: (BC)/(( CAB)) = (AC)/(( ABC)) (35.3)/( 51.6^) = (29.5)/(( ABC)) Step 2: Solve for ( ABC). Cross-multiply to isolate ( ABC): 35.3 × ( ABC) = 29.5 × 51.6^ ( ABC) = (29.5 × 51.6^)/(35.3) ( ABC) = (29.5 × 0.783636...)/(35.3) ( ABC) = (23.11726...)/(35.3) ( ABC) = 0.65488... Step 3: Calculate ABC. ABC = ^-1(0.65488...) ABC ≈ 40.900^ Rounding to one decimal place: ABC = 40.9^ b) Calculate the area of triangle ABC. Step 1: Use the formula for the area of a triangle given two sides and the included angle. The area of a triangle can be calculated using the formula: Area = (1)/(2)ab C where a and b are the lengths of two sides, and C is the included angle between those sides. In ABC, we can use sides AC and AB and the included angle CAB: Side AC = 29.5 cm Side AB = 45 cm Included angle CAB = 51.6^ Step 2: Substitute the values into the formula and calculate the area. Area = (1)/(2) × AC × AB × ( CAB) Area = (1)/(2) × 29.5 cm × 45 cm × (51.6^) Area = (1)/(2) × 1327.5 cm^2 × 0.783636... Area = 663.75 cm^2 × 0.783636... Area ≈ 519.063 cm^2 Rounding to three significant figures: Area = 519 cm^2 3 done, 2 left today. You're making progress.