求解这道题---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 $\angle ACB = 55^\circ$. Find the height of the mountain.
[Diagram labeled Figure 4 (a)]
Solve the above textbook question. (2 marks)
[Handwritten calculation beside Figure 4 (a)]
tan $55^\circ$ = AB/400
AB = 400 tan $55^\circ$
AB = 571 cm.
(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 $\angle ACB$.
II. Move back by x m to point D and measure $\angle ADB$.
[Diagram labeled Figure 4 (b)]
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 = $\frac{x \tan a \tan b}{\tan a - \tan b}$. Do you agree with him? Explain your answer. (5 marks)
(c) Clinton accepted Andy's suggestion and measured that a = $45^\circ$, b = $40^\circ$ and x = 100. By using Andy's formula in (b), find the value of h. (2 marks)
Ans: (a) 571 m ; (c) 522
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**Chart/Diagram Description:**
**Figure 4 (a):**
* Type: Geometric figure representing a right-angled triangle.
* Elements:
* Points: A, B, C.
* Lines: Straight lines form a triangle ABC. AB is a vertical line representing the height. BC is a horizontal line. AC is the hypotenuse. There is a small rectangle at B indicating a right angle at B (assuming the mountain's height is perpendicular to the base).
* Angles: Angle at C is labeled as $55^\circ$. Angle at B is implied to be $90^\circ$. Angle at A is not labeled with a value.
* Lengths: BC is labeled as 400 m. AB is labeled as AB. AC is not labeled.
* Labels: A is at the top (peak of mountain representation). B is at the bottom right. C is at the bottom left.
* Representation: AB is depicted as the height of a mountain profile drawn to the right of AB.
**Figure 4 (b):**
* Type: Geometric figure representing two nested right-angled triangles sharing a common vertical side.
* Elements:
* Points: A, B, C, D. Points D, C, B lie on a horizontal straight line (implied by the text "BCD is a straight line").
* Lines: Straight lines form two right-angled triangles, ABD and ABC. AB is a vertical line representing the height. DB is a horizontal line. DC, CB, AB, AD, AC are lines. A vertical dashed line segment from A down to point B. A horizontal dashed line segment from A to some point above C.
* Angles: Angle at D ($\angle ADB$) is labeled as 'b'. Angle at C ($\angle ACB$) is labeled as 'a'. Angle at B is implied to be $90^\circ$ (height is perpendicular to the base).
* Lengths: DC is labeled as 'x m'. CB is labeled as 'b' (lowercase, which seems inconsistent with the angle labels 'a' and 'b' and the text description which uses 'BC' for the length). Let's assume the diagram label 'b' under BC represents the length BC. The height AB is labeled as 'h'.
* Labels: A is at the top (peak of mountain representation). B is at the bottom directly below A. C is on the horizontal line between D and B. D is on the horizontal line to the left of C.
* Representation: AB is depicted as the height of a mountain profile drawn to the right of AB.
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**Extracted Text:**
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 $\angle ACB = 55^\circ$. Find the height of the mountain.
Solve the above textbook question. (2 marks)
(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 $\angle ACB$.
II. Move back by x m to point D and measure $\angle ADB$.
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 = $\frac{x \tan a \tan b}{\tan a - \tan b}$. Do you agree with him? Explain your answer. (5 marks)
(c) Clinton accepted Andy's suggestion and measured that a = $45^\circ$, b = $40^\circ$ and x = 100. By using Andy's formula in (b), find the value of h. (2 marks)
Ans: (a) 571 m ; (c) 522
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**Handwritten Calculation (for part a):**
tan $55^\circ$ = AB/400
AB = 400 tan $55^\circ$
AB = 571 cm. (Note: This result is given in cm, but the given answer at the bottom says 571 m. There might be a unit inconsistency in the handwritten calculation or the provided answer).
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答案文本
视频字幕
Let's solve part (a) of this mountain height problem. We have a right triangle where AB represents the mountain height, BC equals 400 meters, and angle ACB is 55 degrees. Using trigonometry, we apply the tangent ratio: tan of 55 degrees equals AB over 400. Solving for AB, we get AB equals 400 times tan of 55 degrees, which equals approximately 571 meters.
Now let's examine Andy's practical method in part (b). Andy proposes measuring angle ACB from point C, then moving back x meters to point D and measuring angle ADB. He claims the height h can be found using his formula. Let's verify this. From triangle ABC, tan of a equals h over BC. From triangle ABD, tan of b equals h over BC plus x. Solving these two equations simultaneously, we can derive Andy's formula, confirming that his method is mathematically correct.
Let's work through the mathematical derivation step by step. From triangle ABC, we have tan of a equals h over BC, so BC equals h over tan of a. From triangle ABD, tan of b equals h over BC plus x. Substituting our expression for BC, we get tan of b equals h over h divided by tan of a plus x. Cross multiplying and rearranging terms, we arrive at Andy's formula: h equals x times tan of a times tan of b, all divided by tan of a minus tan of b. This confirms that Andy's formula is mathematically correct.
Now let's solve part (c) using Clinton's measurements. We have a equals 45 degrees, b equals 40 degrees, and x equals 100 meters. Using Andy's formula, we substitute these values. Tan of 45 degrees equals 1, and tan of 40 degrees equals approximately 0.839. Plugging these into the formula: h equals 100 times 1 times 0.839, divided by 1 minus 0.839. This gives us 83.9 divided by 0.161, which equals approximately 521.1 meters. Rounding to the nearest meter, the height is 522 meters.
Let's summarize our complete solution. In part (a), using the traditional trigonometric method, we found the mountain height to be 571 meters. In part (b), we verified that Andy's formula is mathematically correct and provides a practical solution for real-world measurements where direct distance measurement is impossible. In part (c), applying Andy's method with the given measurements, we calculated the height as 522 meters. Andy's approach demonstrates how mathematical theory can be adapted to solve practical surveying challenges.