Problem 315 - Calculus Volume 3

Question

A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.

Step-by-Step Solution

Answer

The depth of the searchlight is 0.5625 feet.

Step 1: Understand the Shape of the Searchlight

The searchlight is a paraboloid of revolution, meaning it is a three-dimensional shape created by rotating a parabola around its axis of symmetry. The key point of interest is the focus, where the light source is placed. This focus is located 1 foot from the base along the axis.

Step 2: Identify the Parabola Properties

The parabola's vertex will be at the origin of a coordinate system placed such that the vertex is at \((0,0)\), and the parabola opens upwards. Since the searchlight opens 3 feet across, the diameter is 3 feet, which gives a radius \(r = \frac{3}{2} = 1.5\) feet.

Step 3: Use the Parabola Equation

For a parabola with a vertex at the origin, the equation is \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus. In this case, since the focus is 1 foot from the vertex, \(p = 1\). Therefore, the equation becomes \(x^2 = 4y\).

Step 4: Determine the Depth of the Paraboloid

The point at which the parabola is 1.5 feet from the center along the x-axis corresponds to half the width across the searchlight. Substitute \(x = 1.5\) into the parabola equation to find \(y\), the depth of the searchlight.\[x^2 = 4y\] Plug in \(x = 1.5\) to give:\[1.5^2 = 4y\] Simplifying gives \[2.25 = 4y\] \[y = \frac{2.25}{4} = 0.5625\] feet.

Key Concepts

Parabola Properties

The parabola is a fundamental curve in mathematics that has several important properties. One of its key qualities is its symmetrical nature. When you draw a line through the vertex (the highest or lowest point of the parabola), each side of the curve mirrors the other. This symmetry along the axis of the parabola makes it a valuable tool in designing objects like searchlights.

• **Symmetry:** Parabolas are symmetrical around their axis.
• **U-shape:** The typical visual form, either opening upwards or downwards.
• **Directrix and Focus:** These are unique references that define the properties of a parabola.

Parabolas are often used in various real-world applications, from satellite dishes to suspension bridges, showcasing their versatility and functionality. Utilizing these properties allows engineers to precisely control the direction and area over which light is projected, especially in the case of a searchlight designed as a paraboloid of revolution.

Focus and Vertex

The focus and vertex are crucial points in understanding a parabola's structure. The vertex is the midpoint and the point of symmetry, while the focus is a point that lies inside the parabola connecting via a straight line perpendicular to the directrix.

• **Vertex:** Often used as the origin in a coordinate system for simplified calculations.
• **Focus:** Determines how light or sound waves are reflected within a parabolic shape.

In the context of a searchlight, the focus is where the light source is placed to ensure that rays reflecting off the paraboloid surface are emitted in a uniform direction. By positioning the light at the focus, the beam can be directed with precision, illuminating objects effectively at a distance.

Parabola Equation

The equation of a parabola helps define its precise shape on a coordinate plane. One common form is \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus. This equation is utilized when the vertex of the parabola is positioned at the origin of the coordinate system, and it opens along the y-axis.

• **Standard form:** \(x^2 = 4py\) for vertical parabolas.
• **Parameter \(p\):** Represents the focal length or the distance from the vertex to focus.

For the searchlight example, since the focus is positioned 1 foot away from the vertex, the parabola's formula becomes \(x^2 = 4y\). This equation serves as a blueprint, helping to calculate the specific dimensions needed for optimal light projection.

Coordinate System in Calculus

A coordinate system is an essential tool in calculus, providing a framework for graphically representing mathematical relations and functions. In the study of parabolas, a two-dimensional Cartesian coordinate system is typically used, where the x-axis and y-axis intersect perpendicularly.

• **Cartesian Plane:** Defined by a pair of perpendicular lines (axes).
• **Origin:** The point where the axes meet, usually assigned zero values for both coordinates.
• **Axis of symmetry:** Vertically or horizontally through the vertex for parabolas.

This system allows us to place the vertex of a parabola at the origin, simplifying calculations and visualizations. By using this setup, geometric properties and dimensions such as the depth of the paraboloid can be easily derived. It is crucial in bridging the abstract concepts of calculus with practical applications like designing searchlights.