Math Problem Statement

Two observers who are 5 km apart simultaneously sight a

small airplane flying between them. One observer measures a

51.0-degree inclination angle, while the other observer measures a

40.5-degree inclination angle, as shown in the diagram. At what

altitude is the airplane flying? just give ans

Solution

To find the altitude of the airplane, use the following formula derived from trigonometry:

Altitude=d⋅tan⁡(θ1)⋅tan⁡(θ2)tan⁡(θ1)+tan⁡(θ2)\text{Altitude} = \frac{d \cdot \tan(\theta_1) \cdot \tan(\theta_2)}{\tan(\theta_1) + \tan(\theta_2)}

Where:

  • d=5 kmd = 5 \, \text{km} (distance between observers),
  • θ1=51.0∘\theta_1 = 51.0^\circ,
  • θ2=40.5∘\theta_2 = 40.5^\circ.

Answer: The altitude of the airplane is approximately 1.73 km.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle Properties
Tangent Function

Formulas

Altitude = (d * tan(θ1) * tan(θ2)) / (tan(θ1) + tan(θ2))

Theorems

Tangent Ratio in Right Triangles

Suitable Grade Level

Grades 10-12

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