Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 3 feet across at the opening and 1 foot deep, where is the focus?

Step-by-Step Solution

Answer

The focus is \( \frac{9}{16} \) feet from the vertex.

Step 1: Define the Paraboloid Equation

A paraboloid can be represented by the equation \( y = ax^2 \) where the vertex is at the origin. The paraboloid opens upwards, so we use this standard form to model it since the reflector has a circular cross-section.

Step 2: Use Given Dimensions

Given that the paraboloid is 3 feet across, the diameter of the opening is 3 feet. This means from the center (vertex) to the edge is 1.5 feet. The depth, or the distance from the vertex to the opening along the y-axis, is 1 foot.

Step 3: Substitute Values to Find 'a'

Using the point at the edge of the reflector (1.5, 1) in the equation \( y = ax^2 \), substitute \( y = 1 \) and \( x = 1.5 \) to find \( a \). So, \( 1 = a(1.5)^2 \). Solving this, \( a = \frac{4}{9} \).

Step 4: Determine the Focus

The focus of a paraboloid with an equation \( y = ax^2 \) is located at \( y = \frac{1}{4a} \). Substitute \( a = \frac{4}{9} \) into the equation to find the focus: \( y = \frac{1}{4 \times \frac{4}{9}} = \frac{9}{16} \) feet.

Key Concepts

Focus of a Paraboloid

In the context of a paraboloid, the **focus** is a crucial point where reflected rays parallel to the axis of symmetry will converge. This property makes paraboloids ideal shapes for devices like searchlights or telescopes, where directing or collecting light is essential. In our exercise, the focus is the point where the light source should be placed to ensure that light beams are reflected evenly across the circular cross-section of the paraboloid reflector. Understanding how to locate the focus involves knowing the parabola's equation, as it is directly tied to the coefficient of the equation. The formula for the focus of a paraboloid represented by the equation \( y = ax^2 \) is \( y = \frac{1}{4a} \). This provides the y-coordinate of the focus when the paraboloid apex or vertex is at the origin.

The Reflector

A reflector is designed to project or collect light rays to or from a particular direction. In the case of a paraboloid reflector, the shape serves a dual purpose: gathering rays entering parallel to the paraboloid's axis and reflecting them through the focus, or projecting rays from the focus outward in parallel lines. This unique quality stems from the parabolic shape that ensures every ray reflecting off the surface does so at an equal angle relative to the focus, defined by the laws of reflection. Reflectors are commonly used in practical devices like headlights, streetlights, and satellite dishes due to this consistent directing capability.

Vertex of a Paraboloid

The **vertex** of a paraboloid is the pivot point where the paraboloid's u-shape begins. It is the point that defines the minimum or maximum value of the parabola, depending on the orientation. In our case, with the paraboloid equation \( y = ax^2 \), which opens upwards, the vertex is at the origin (0,0). This point is central to calculating other relevant outputs such as the focus; knowing the vertex location simplifies setting up the geometric scheme needed for further mathematical calculations. It's also the point through which the paraboloid's axis of symmetry passes, making it an anchor point for tasks like aligning a reflector.

Equation of a Paraboloid

The **equation** of a paraboloid lays the foundation for understanding its geometric and reflective properties. Typically expressed as \( y = ax^2 \), this formula defines a u-shaped curve with its vertex at the origin. The constant \( a \) determines how "open" or "narrow" the parabola appears. A larger \( a \) value results in a narrower parabola. In exercises like ours, substituting known values into the equation helps determine \( a \), allowing us to calculate critical features like the location of the focus. Every point \((x, y)\) on this curve represents potential positions in the physical setup, which becomes crucial when designing or analyzing the effectiveness of reflectors.