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An observer in a hot-air balloon sees a building that is 50m away. The balloon has a height of 165m. What is the angle of depression from the balloon to the building?

An observer in a hot-air balloon sees a building that is 50 m 50 \mathrm{~m} away. The balloon has a height of 165 m 165 \mathrm{~m} .\newlineWhat is the angle of depression from the balloon to the building?

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Full solution

Q. An observer in a hot-air balloon sees a building that is 50 m 50 \mathrm{~m} away. The balloon has a height of 165 m 165 \mathrm{~m} .\newlineWhat is the angle of depression from the balloon to the building?
  1. Understand scenario and triangle: Understand the scenario and the triangle involved.\newlineThe observer is in a hot-air balloon directly above a point that is 50 m50\,\text{m} horizontally from the building. The height of the balloon is 165 m165\,\text{m}. This forms a right-angled triangle with the building, the point on the ground directly below the balloon, and the balloon itself.
  2. Identify sides of triangle: Identify the sides of the triangle.\newlineThe height of the balloon is the opposite side of the right-angled triangle, which is 165 m165\,\text{m}. The distance from the building to the point on the ground directly below the balloon is the adjacent side, which is 50 m50\,\text{m}.
  3. Use tangent function: Use the tangent function to find the angle of depression.\newlineThe tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. Therefore, we can use the formula:\newlinetan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  4. Calculate angle with arctangent: Calculate the angle using the arctangent function.\newlinetan⁡(θ)=165m50m\tan(\theta) = \frac{165m}{50m}\newlineθ=arctan⁡(16550)\theta = \arctan\left(\frac{165}{50}\right)\newlineθ≈arctan⁡(3.3)\theta \approx \arctan(3.3)
  5. Use calculator to find angle: Use a calculator to find the angle of depression. \newlineθ≈arctan⁡(3.3)\theta \approx \arctan(3.3)\newlineθ≈73.3\theta \approx 73.3 degrees (using a calculator to find the arctangent of 3.33.3)

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