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A reflecting telescope uses a mirror shaped like a paraboloid of revolution. This shape is created by rotating a parabola around its axis. It is widely used because it directs incoming light rays to a single point, known as the focus.

In the exercise, the mirror has a diameter (or width) of 4 inches and a depth of 3 inches. When rotating a parabola to form this shape, it creates a 3D object. This unique design helps in gathering light and improving image quality.

To understand how the mirror collects light, we need to delve into the core geometry of the paraboloid.

The equation of a parabola is fundamental in solving this problem. It looks like this in its standard form:

\(y = ax^2\)

Here, 'y' is the depth, and 'x' represents the horizontal distance from the central axis of the parabola.

The mirror described in the problem has a diameter of 4 inches, therefore, the radius is 2 inches. The depth of the mirror is given as 3 inches. By positioning the vertex of the parabola at the origin (0, 0), this means when \(x=2, y=3\).

To find the specific value of 'a', we substitute the known values into the equation, \(3 = a(2)^2\), and solve for 'a':

\(a = \frac{3}{4}\).

Now, the equation of our specific parabola becomes \(y = \frac{3}{4}x^2\).

Understanding the geometry of the mirror is crucial for solving where the light will be concentrated. Since the mirror is a paraboloid of revolution, each cross-section parallel to the axis of the parabola is a parabola.

The light rays parallel to the axis reflect off the surface and converge at a point known as the focal point. This property makes paraboloids highly efficient in focusing light.

When light hits the mirror, it reflects following the rule: the angle of incidence equals the angle of reflection. In a paraboloid, this reflection results in light rays converging at one specific point, making it ideal for telescopes.

Finding the focus of a parabola helps determine where the light collected by the telescope mirror will be concentrated. For a parabola of the form \(y = ax^2\), the distance from the vertex to the focus is given by:

\[\frac{1}{4a}\].

With 'a' calculated earlier as \(\frac{3}{4}\), we substitute it into the focus formula:
\[\frac{1}{4 \times \frac{3}{4}} = \frac{1}{3}\].

The focus, therefore, lies \( \frac{1}{3} \) inches above the vertex.

In conclusion, the reflecting mirror in this telescope will concentrate all collected light at a point 1 inch above the vertex, making it an efficient design for focusing incoming light rays.