Observers at positions A and B 2 km apart simultaneously measure the angle of elevation of a weather balloon to be 40° and 70°, respectively. If the balloon is directly above a point on the line segment between A and B, find the height of the balloon.

검증된 단계별 안내

First, visualize the problem by drawing a diagram. Place points A and B on a horizontal line, 2 km apart. Let point C be the point directly below the balloon on the line segment AB. The balloon is at point D, directly above C. You have two right triangles: ACD and BCD.

In triangle ACD, angle A is 40°. Use the tangent function, which relates the angle of elevation to the opposite side (height of the balloon, h) and the adjacent side (distance from A to C, x). The equation is: tan(40°) = h / x.

In triangle BCD, angle B is 70°. Similarly, use the tangent function: tan(70°) = h / (2 - x), where (2 - x) is the distance from B to C.

You now have two equations: tan(40°) = h / x and tan(70°) = h / (2 - x). Set these equations equal to each other to eliminate h: tan(40°) * x = tan(70°) * (2 - x).

Solve the equation for x, which represents the distance from A to C. Once you have x, substitute it back into either equation (e.g., h = x * tan(40°)) to find the height of the balloon, h.

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주요 개념

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this problem, the tangent function is particularly useful, as it connects the angle of elevation to the height of the balloon and the horizontal distance from the observers to the point directly below the balloon.

Right Triangle Geometry

The scenario involves right triangles formed by the observers, the height of the balloon, and the horizontal distances from the observers to the point directly below the balloon. Understanding the properties of right triangles, including the Pythagorean theorem, is essential for calculating unknown lengths and heights based on the angles provided.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line from an observer's eye to the object being observed and the line of sight to that object. In this problem, the angles of elevation from points A and B to the balloon allow us to set up equations that relate the height of the balloon to the distances from the observers, enabling us to solve for the height.