Problem 52 - Algebra and Trigonometry

Question

The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.

Step-by-Step Solution

Answer

The focal length is approximately 659.37 inches.

Step 1: Understand the Problem

We need to find the focal length of the parabolic mirror given its diameter and depth. The diameter of the mirror is 200 inches, and it is 3.79 inches deep at the center. We assume the standard form of the parabola for the mirror is \( y = ax^2 \).

Step 2: Establish the Parabolic Equation

For a parabolic mirror, the vertex of the parabola is at the center. With the vertex at the origin, the equation can be written as \( x^2 = 4py \) where \( p \) is the focal length. Here, \( y \) represents the depth of the parabola at any point \( x \). Since the maximum depth of the mirror is 3.79 inches, we translate this into our equation: when \( y = 3.79 \), \( x = 100 \) because the radius (half of the diameter) is 100 inches.

Step 3: Solve the Parabolic Equation for Focal Length

Substitute the known values into the equation \( x^2 = 4py \):\[ 100^2 = 4p imes 3.79 \]This simplifies to:\[ 10000 = 15.16p \]Solve for \( p \):\[ p = \frac{10000}{15.16} \]

Step 4: Calculate and Conclude

Perform the division:\[ p = \frac{10000}{15.16} \approx 659.37 \]So, the focal length of the parabolic mirror is approximately 659.37 inches.

Key Concepts

Parabola Equation

A parabola is a symmetrical curve that can be defined by a specific equation. For a parabolic mirror, which is designed to focus light, the equation is typically set in the standard form: \( y = ax^2 \), where \( a \) is a constant that determines the width and direction of the parabola. In our context, this equation reflects how the mirror's surface is shaped to collect and focus light.

The mirror's form can also be expressed using a different form, \( x^2 = 4py \), where \( p \) is the focal length. This form is more directly useful for calculating the focal length of the mirror. Each symbol in the equation has specific meanings:

• \( x \) represents the distance from the vertex horizontally across the parabola.
• \( y \) denotes the depth of the parabola at any point \( x \).

The value of \( p \), the focal length, is crucial to finding how well the mirror focuses light, and this can be found by manipulating this equation.

Vertex of Parabola

The vertex of a parabola is a significant point. It is the tip of the parabola where the curve changes direction. For mirrors shaped like a parabola, the vertex is usually positioned at the deepest point of the curve. In this exercise, the vertex is at the center of the mirror and acts as a crucial reference point for the mirror's geometry.

When we set up our parabola equation as \( x^2 = 4py \), we assume the vertex is at the origin \((0,0)\). This makes calculations much simpler. Here, \( x = 0 \) and \( y = 0 \) mean the vertex is right in the middle horizontally and vertically at the base.

The vertex helps us derive the equation by giving us a starting point and ensuring that our measurements, particularly in terms of depth (\( y \)), are from this central position. Understanding the position of the vertex within the mirror design allows for precise calculations of focal length.

Focal Length Calculation

The focal length of a parabolic mirror is the distance from the vertex to the focus, where all parallel rays of light converge. This calculation is pivotal in understanding how the mirror focuses light effectively.

To find the focal length \( p \), we begin by taking a specific set of measurements. First, using the equation \( x^2 = 4py \), we substitute known values: when \( x = 100 \) inches, representing half the mirror's diameter, \( y = 3.79 \), the depth of the mirror.

Substituting these values gives us:
\[100^2 = 4p \times 3.79 \]
Simplifying, we have \( 10000 = 15.16p \). Solving this results in:
\[ p = \frac{10000}{15.16} \approx 659.37 \]
The calculated focal length is approximately 659.37 inches. This tells us where incoming parallel light rays converge, thus defining how the mirror effectively focuses light.

Mirror Diameter and Depth

Understanding how mirror diameter and depth affect focal length is key in measuring and designing parabolic mirrors. The diameter of a mirror is the longest distance across it, and for a circular mirror, it's simply twice the radius. For our parabolic mirror, a diameter of 200 inches translates into a radius of 100 inches, which will inform our equation.

The depth of a mirror is the distance from the vertex to its shallowest point (edge), and for the Hale telescope’s mirror, it is 3.79 inches. This depth represents \( y \) in the parabola equation \( x^2 = 4py \).

These dimensions directly influence the mirror's focal length by providing the necessary measurements to plug into the equation. A deeper or wider mirror will refract and focus light differently. Understanding these factors is crucial when calculating focal points and designing mirrors to achieve desired optical effects.