Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

검증된 단계별 안내

Identify the right triangles formed by the wires and the pole. Let the height of the pole be h, the distance from the base of the pole to point B be x, and the distance from the base of the pole to point C be x - 10 m.

Use trigonometry to express the height of the pole in terms of x. For triangle T-B, use the tangent function: \( \tan(35^\circ) = \frac{h}{x} \). This gives the equation: \( h = x \cdot \tan(35^\circ) \).

Similarly, for triangle T-C, use the tangent function: \( \tan(50^\circ) = \frac{h}{x - 10} \). This gives the equation: \( h = (x - 10) \cdot \tan(50^\circ) \).

Set the two expressions for h equal to each other: \( x \cdot \tan(35^\circ) = (x - 10) \cdot \tan(50^\circ) \).

Solve the equation for x, and then substitute back into either expression for h to find the height of the pole.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trigonometric Ratios

Trigonometric ratios relate the angles and sides of a right triangle. In this problem, the sine, cosine, and tangent functions will be used to find the height of the pole based on the angles formed by the wires with the horizontal. For instance, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, which is crucial for determining the height of the pole.

Right Triangle Properties

The problem involves right triangles formed by the vertical pole and the wires. Each wire creates a right triangle with the pole as one side and the horizontal distance to points B and C as the other side. Understanding the properties of right triangles, including the Pythagorean theorem, is essential for calculating the height of the pole based on the lengths of the sides and the angles given.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal. In this scenario, the angles of elevation from points B and C to the top of the pole are given, which allows us to set up equations using trigonometric functions to find the height of the pole. Recognizing how these angles relate to the height and distances involved is key to solving the problem.