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 how to solve the problem: Step 1: Identify the given information. Let the helicopter's position be C, and the two points on the horizontal plane be A and B. We are given: • Distance from helicopter to A, AC = 850 m • Distance from helicopter to B, BC = 1200 m • Angle of elevation from B to the helicopter, B = 43^ We need to find the distance AB. Let AB = x. Step 2: Apply the Law of Cosines. The Law of Cosines states a^2 = b^2 + c^2 - 2bc A. In our triangle ABC, we can write it as: AC^2 = AB^2 + BC^2 - 2(AB)(BC) B Substitute the known values: 850^2 = x^2 + 1200^2 - 2(x)(1200) 43^ 722500 = x^2 + 1440000 - 2400x 43^ Step 3: Solve the quadratic equation. First, calculate 43^: 43^ ≈ 0.7313537 Substitute this value into the equation: 722500 = x^2 + 1440000 - 2400x (0.7313537) 722500 = x^2 + 1440000 - 1755.24888x Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0: x^2 - 1755.24888x + (1440000 - 722500) = 0 x^2 - 1755.24888x + 717500 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = 1755.24888 ± sqrt((-1755.24888)^2 - 4(1)(717500))2(1) x = 1755.24888 ± sqrt(3080898.10 - 2870000)2 x = 1755.24888 ± sqrt(210898.10)2 x = (1755.24888 ± 459.23643)/(2) This gives two possible values for x: x_1 = (1755.24888 + 459.23643)/(2) = (2214.48531)/(2) ≈ 1107.24 m x_2 = (1755.24888 - 459.23643)/(2) = (1296.01245)/(2) ≈ 648.01 m Step 4: Interpret the solutions. This is an ambiguous case (SSA) where two valid triangles can be formed. • If AB ≈ 1107.24 m, then A ≈ 74.3^ (acute). • If AB ≈ 648.01 m, then A ≈ 105.7^ (obtuse). The diagram provided shows A as an acute angle. Therefore, we select the solution that corresponds to an acute A. The distance AB is approximately 1107.2 m. The final answer is 1107.2 m. Send me the next one 📸