The shadow cast by a lampshade has a boundary in the shape of a hyperbola defined by the equation (y−3)22−x21=1
. What could the center of the hyperbola represent? a. the location of the light bulb b. the bottom of the lampshade c. the top of the lampshade d. a vertex of the shadow
The Correct Answer and Explanation is:
Correct Answer: a. the location of the light bulb
Explanation:
The given equation of the hyperbola is: (y−3)222−x212=1\frac{(y – 3)^2}{2^2} – \frac{x^2}{1^2} = 122(y−3)2−12×2=1
This is in the standard form of a vertical hyperbola: (y−k)2a2−(x−h)2b2=1\frac{(y – k)^2}{a^2} – \frac{(x – h)^2}{b^2} = 1a2(y−k)2−b2(x−h)2=1
From this, we can identify the center of the hyperbola as the point (h, k). Comparing the equation:
- h=0h = 0h=0
- k=3k = 3k=3
So, the center is at the point (0, 3).
In the context of a lampshade casting a shadow, the shape of the shadow being a hyperbola suggests that the light is shining through or around an object and creating that curved boundary on a surface. The center of a hyperbola represents the midpoint between the two branches and also aligns with the origin of the diverging light rays that create the shadow. This is consistent with how a point source of light, such as a light bulb, causes objects to cast shadows with curved edges when blocked or filtered by another object like a lampshade.
Since the center of this hyperbola is (0, 3), it most likely indicates the position from which the light rays originate. This is exactly what a light bulb represents in a real-world setting.
The other answer choices do not align with the mathematical interpretation:
- The bottom or top of the lampshade (choices b and c) are not directly linked to the center of the hyperbola
- A vertex (choice d) lies along the axis but is not the center
Therefore, the most accurate interpretation is that the center of the hyperbola represents the location of the light bulb.
