Digital SATMath: Geometry and TrigonometryHard
A satellite dish is designed as a parabolic reflector. The cross-section of the dish can be modeled by the equation y = (1/16)x², where x and y are in meters. What is the focal length of the parabolic dish?
- A4 meters
- B8 meters
- C16 meters
- D32 meters
Show answer & explanationAnswer & explanation
Correct answer: A. 4 meters
The standard equation for a parabola opening upwards or downwards with its vertex at the origin is x² = 4py or y = (1/(4p))x², where 'p' is the focal length. Comparing the given equation y = (1/16)x² to the standard form, we have 1/(4p) = 1/16. This means 4p = 16. Solving for p, we get p = 16/4 = 4 meters. The focal length is 4 meters.
Why the other options are wrong
- B. This might come from incorrectly using 1/(2p) or similar error.
- C. This is the denominator '16' directly, without dividing by 4.
- D. This is 2 * 16, or a significant miscalculation of the '4p' relationship.
Focal Length of a Parabola
The distance from the vertex to the focus (or directrix) of a parabola.
- For y = (1/(4p))x² (vertex at origin, opens up/down), 'p' is the focal length.
- The focus is a key point in optical and acoustic applications (e.g., satellite dishes).
- Determines the 'width' or 'depth' of the parabolic curve.
Memory trick: A parabola's focus is where parallel rays meet, like a spotlight.