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Question 1175022: A flashlight shape like a paraboloid has its light source at the focus located 1.5cm from the base and is 10cm wide at its opening. How deep is the flashlight at its center?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Understand the Paraboloid**
* A paraboloid is a 3D shape formed by rotating a parabola around its axis of symmetry.
* The focus is a point inside the paraboloid where all reflected light rays converge.
* The distance from the vertex (the bottom point) of the paraboloid to the focus is the focal length.
**2. Set Up the Parabola**
* We can represent the cross-section of the flashlight as a parabola in a coordinate system.
* Let the vertex of the parabola be at the origin (0, 0).
* The focus is at (0, 1.5), meaning the focal length (p) is 1.5 cm.
* The parabola opens upward, so its equation is x² = 4py.
**3. Use the Given Dimensions**
* The flashlight is 10 cm wide at the opening, so the radius is 5 cm.
* This means when x = 5, we need to find the corresponding y-value, which represents the depth.
**4. Calculate the Depth**
* Substitute the focal length (p = 1.5) into the equation: x² = 4(1.5)y, which simplifies to x² = 6y.
* Substitute x = 5 into this equation: 5² = 6y, so 25 = 6y.
* Solve for y: y = 25 / 6 ≈ 4.1667 cm.
**Answer:**
The flashlight is approximately 4.17 cm deep at its center.
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