Question 1175022
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.