Digital SATMath: Geometry and TrigonometryHard
An engineer is designing a spiral staircase. Each step rises 8 inches and has a tread length of 15 inches. If the staircase forms a continuous slope, what is the angle of elevation of the staircase to the nearest degree?
- A32°
- B45°
- C62°
- D28°
Show answer & explanationAnswer & explanation
Correct answer: D. 28°
Each step forms a right triangle where the rise (8 inches) is the opposite side and the tread (15 inches) is the adjacent side to the angle of elevation. We use the tangent function: tan(θ) = opposite/adjacent = 8/15. To find the angle (θ), we use the inverse tangent function: θ = arctan(8/15). 8/15 ≈ 0.5333. arctan(0.5333) ≈ 28.07°. Rounded to the nearest degree, the angle is 28°.
Why the other options are wrong
- A. This is a plausible distractor from incorrect calculation or rounding.
- B. This would imply equal rise and tread, which is not the case.
- C. This would result from using 15/8 instead of 8/15, or using arcsin/arccos incorrectly.
Inverse Tangent (Arctan)
The inverse trigonometric function that finds the angle whose tangent is a given ratio.
- Denoted as tan⁻¹ or arctan.
- Used to find an unknown angle in a right triangle when opposite and adjacent sides are known.
- Result is typically in degrees or radians.
Memory trick: To find an angle, you need the 'inverse' function – like unwrapping a present.