求解---2013 C4 (Modified) (a) A textbook question reads as follows: In Figure 4 (a), suppose AB represents the height of a mountain. It is measured that BC = 400 m and ∠ACB = 55°. Find the height of the mountain. Solve the above textbook question. (2 marks) Handwritten notes next to Figure 4(a): tan 55° = AB / 400 AB = tan 55° * 400 AB = 571 cm. (Note: The unit 'cm' appears handwritten, but given the context of a mountain and 400m base, 'm' is expected). Figure 4 (a) Description: Geometric figure: Right-angled triangle ABC, with angle B being the right angle. A is at the top, B is at the base directly below A, and C is a point on the ground. AB represents the height of the mountain. Labels: A, B, C. Lengths: BC = 400 m. Angles: ∠ACB = 55°. Angle at B is a right angle. Annotation: AB represents the height. Figure Label: Figure 4 (a). (b) Andy comments, 'As part of BC lies at the bottom of the mountain, it is impossible to measure the length of BC by using the simplest measuring instruments. Although the question can be solved, it is impractical to do so.' Hence, he proposes the following way to measure the height of the mountain (see Figure 4 (b)). I. Use a protractor to measure ∠ACB. II. Move back by x m to point D and measure ∠ADB. Suppose BCD is a straight line. With the notations in Figure 4 (b), Andy claims that the height of the mountain h m can be found by using the formula h = x tan a tan b / tan a - tan b. Do you agree with him? Explain your answer. (5 marks) Figure 4 (b) Description: Geometric figure: Two right-angled triangles, ABC and ABD, sharing a common vertical side AB. Points D, C, B lie on a straight horizontal line. B is the vertex with the right angle. A is at the top. D is further away from B than C. Labels: A, B, C, D. Lengths: AB = h m (height). DC = x m. BC and BD are horizontal distances from the base B to points C and D respectively. Angles: ∠ACB = a, ∠ADB = b. Angle at B is a right angle. Annotation: BCD is a straight line. Figure Label: Figure 4 (b). (c) Clinton accepted Andy's suggestion and measured that a = 45°, b = 40° and x = 100. By using Andy's formula in (b), find the value of h. (2 marks) Answer provided: Ans: (a) 571 m; (c) 522

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Let's solve part (a) of this mountain height problem. We have a right triangle ABC where AB represents the mountain height, BC equals 400 meters, and angle ACB equals 55 degrees. Using the tangent ratio, tan 55 degrees equals AB over BC. Therefore, AB equals 400 times tan 55 degrees, which gives us approximately 571 meters. Now let's examine part (b). Andy proposes a more practical method to measure the mountain height. Instead of measuring the base distance BC directly, he suggests measuring two angles from different positions. First, measure angle ACB equals a from point C. Then move back x meters to point D and measure angle ADB equals b. Andy claims the height h can be found using the formula: h equals x times tan a times tan b, divided by tan a minus tan b. Let's verify this formula by deriving it from basic trigonometry. Let's derive Andy's formula step by step. In triangle ABC, tan a equals h over BC, so BC equals h over tan a. In triangle ABD, tan b equals h over BD. Since BD equals BC plus x, we have tan b equals h over BC plus x. Substituting our expression for BC, we get tan b equals h over h over tan a plus x. Rearranging this equation: h over tan a plus x equals h over tan b. Solving for x: x equals h over tan b minus h over tan a. Factoring out h: x equals h times one over tan b minus one over tan a. This simplifies to x equals h times tan a minus tan b over tan a tan b. Therefore, h equals x tan a tan b over tan a minus tan b. Yes, I agree with Andy's formula! Now let's solve part (c) using Andy's formula. Clinton measured a equals 45 degrees, b equals 40 degrees, and x equals 100 meters. Using Andy's formula: h equals x tan a tan b divided by tan a minus tan b. Substituting the values: h equals 100 times tan 45 degrees times tan 40 degrees, divided by tan 45 degrees minus tan 40 degrees. Since tan 45 degrees equals 1 and tan 40 degrees approximately equals 0.839, we get: h equals 100 times 1 times 0.839, divided by 1 minus 0.839. This gives us h equals 83.9 divided by 0.161, which equals approximately 522 meters. Let's summarize our complete solution to this mountain height problem. For part (a), using the traditional trigonometric method with the given base distance and angle, we found the mountain height to be 571 meters. For part (b), we verified that Andy's proposed formula is mathematically correct through step-by-step derivation. The formula h equals x tan a tan b divided by tan a minus tan b is indeed valid. For part (c), applying Andy's method with the measured angles of 45 and 40 degrees and a distance of 100 meters, we calculated the height as 522 meters. This problem demonstrates two important approaches to indirect measurement: the traditional method requiring base distance measurement, and Andy's more practical method using only angle measurements from two positions.