Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect? Exercise 43

Step-by-Step Solution

Answer

The incoming light collects 4 inches from the center of the mirror.

Step 1: Understand the Paraboloid Equation

A paraboloid can be described by the equation \(z = \frac{x^2 + y^2}{4f}\), where \(f\) is the focal length of the paraboloid. The cross-section of this paraboloid through its axis is a parabola.

Step 2: Use Known Values

The diameter of the mirror is 8 inches, so the maximum value of \(x\) or \(y\) is 4 inches (radius). The depth of the mirror along the \(z\) axis is 1 inch.

Step 3: Formulate a Parabolic Equation

The cross-section through the axis yields the equation \(z = \frac{x^2}{4f}\). Using the maximum \(x\) value (4 inches) and \(z = 1\) inch, we solve for \(f\): \[1 = \frac{4^2}{4f} \Longrightarrow 1 = \frac{16}{4f} \Longrightarrow f = 4\]

Step 4: Determine the Focal Length

The solution to the previous equation gives us \(f = 4\). This means the focus, or the point where the light collects, is 4 inches from the vertex of the paraboloid.

Key Concepts

Focal Length

The focal length of a paraboloid is a critical measurement that tells us how far away light converges from the vertex. This concept is central to understanding how parabolic mirrors, like those used in reflecting telescopes, work. In simple terms, the focal length ( f ) is the distance between the surface of the mirror at its vertex and the focus, where light focuses after reflecting from the mirror.
In the exercise provided, we derived the focal length using the parabola equation, essential for determining where incoming light rays parallel to the paraboloid's axis will meet. By knowing the diameter and depth of the paraboloid mirror, we find that light converges at a focal length of 4 inches.

Reflecting Telescope Mirror

Reflecting telescopes use mirrors with a parabolic shape to gather and focus light. These mirrors are pivotal because they ensure all parallel incoming light rays converge at the focal point, providing a clear image of distant astronomical objects.
Reflecting telescope mirrors are typically designed as paraboloidal mirrors because of their unique property to accurately focus light. When incoming light rays, which are usually parallel as they arrive from distant stars, hit the parabolic shape, they reflect and meet at the focus point. This is why correctly determining the focal length, as done in our earlier exercise, is so important. It ensures the image formed at the focal plane is sharp, which is vital for astronomers aiming to see clearly into the cosmos.

Parabola Equation

The parabola equation plays a fundamental role in defining the shape and behavior of a paraboloid. The equation for a paraboloid is generally expressed as: \[ z = \frac{x^2 + y^2}{4f} \] where \( f \) is the focal length. This is a three-dimensional form of a parabola, extending the basic two-dimensional concept into a form useful for shaping telescope mirrors.
By taking a cross-section through the axis of a paraboloid, you get the typical parabola equation \( z = \frac{x^2}{4f} \). This simplification allows us to compute important properties of the paraboloid, such as its focal length. In our exercise, understanding this equation helped us solve for the focal length by substituting real-world measurements of the mirror into the equation.

Cross-Section Analysis

Analyzing the cross-section of a paraboloid involves understanding the geometry of the cut through its axis, resulting in a parabola. This is crucial when working with parabolic mirrors, as it provides a two-dimensional representation of a real-world three-dimensional surface.
In the context of our exercise, cross-section analysis allowed us to use the simplified form of the parabola equation, \( z = \frac{x^2}{4f} \), focusing simply on the primary axes (z and x). Using the mirror's given dimensions, we were able to pinpoint the focal point of the paraboloid accurately by calculating \( f \), or focal length. Cross-section analysis simplifies the problem, making it possible to focus only on essential features, critical for engineering applications like telescope mirror design.