Digital SATMath: Geometry and TrigonometryMedium
A civil engineer is designing a new road that needs to go up a hill. The road will rise 15 meters for every 100 meters of horizontal distance. What is the angle of elevation of the road to the nearest degree?
- A8°
- B85°
- C15°
- D9°
Show answer & explanationAnswer & explanation
Correct answer: A. 8°
The rise and run form a right triangle. The angle of elevation can be found using the tangent ratio (opposite/adjacent) and then the inverse tangent function.
Why the other options are wrong
- B. Incorrect. This is a very steep angle, implying a much larger rise for the given run.
- C. Incorrect. This would be the angle if the rise was 100 * tan(15°) = 26.8 meters.
- D. Correct. tan(angle) = 15/100 = 0.15. arctan(0.15) ≈ 8.53°, which rounds to 9°.
Angle of Elevation (Tangent)
The angle of elevation is the angle formed by the horizontal line of sight and the line of sight upwards to an object. It can be found using the tangent ratio (opposite/adjacent) and its inverse function.
- Tangent relates the opposite and adjacent sides of a right triangle to an angle.
- arctan (or tan⁻¹) is used to find the angle when the ratio is known.
- The 'opposite' side is the rise, and the 'adjacent' side is the horizontal distance for angle of elevation.
Memory trick: SOH CAH TOA: Tan for Opposite and Adjacent!