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.

Short answer

The depth of the searchlight is 0.5625 feet.

Step-by-step solution

Understanding the Parabola

A paraboloid of revolution is a three-dimensional shape created by rotating a parabola around its axis of symmetry. The standard equation for the parabola in two dimensions is $y=a{x}^2$, where $a$ is a constant. The light source is located at its focus, $(0,f)$, where $f=1$ foot in this problem.

Determine the Relationship with Width

The width of the searchlight's opening is 3 feet across, which is the distance between the points (-1.5, y) and (1.5, y) on the parabola. Substituting $x=1.5$ into the parabola's equation gives $y=a(1.5{)}^2=2.25a$.

Focus-Vertex Relation

The distance from the vertex to the focus is given by $f=\frac{1}{4a}$. Since the light focus is 1 foot from the base, $1=\frac{1}{4a}$, giving $a=\frac{1}{4}$.

Calculate the Depth

The depth is the $y$-coordinate of the parabola at the points $x=\pm 1.5$. Substituting $a=\frac{1}{4}$ into the depth formula from Step 2, we have $y=2.25\times \frac{1}{4}=0.5625$. Thus, the depth of the searchlight is 0.5625 feet.

Key concepts

Focus of Parabola

The focus of a parabola is a special point that helps define the curve's shape. It is one of the key components of any parabola, along with the directrix. The focus lies on the axis of symmetry of the parabola and is typically denoted as $(0,f)$ if the vertex is at the origin. This focus plays a crucial role in how parabolic shapes direct light or sound, making them ideal for applications like searchlights or satellite dishes.

• Properties: The focus is located inside the parabola.
• For a searchlight, aligning the light source with the focus ensures that light rays reflect parallel from the surface.

Understanding the relationship between the focus and the vertex helps us to manage equations directly affecting the parabola's geometry.

Vertex of Parabola

The vertex of a parabola represents its highest or lowest point, depending on its orientation. In our example, the vertex is the starting point of the shape as it forms the paraboloid of revolution. This specific point is crucial for determining other geometric properties, including symmetry and focal distance.

• The standard form of a parabola in geometry is $y=a{x}^2$.
• The vertex here lies at the origin for simplicity, usually denoted as $(0,0)$.

By considering the vertex’s position, we can determine the vertical or horizontal stretch and direction of the parabola. It's also essential for computing the focal length, which measures from the vertex to the focus.

Equation of Parabola

The equation of a parabola provides a mathematical description of its shape and position. For a parabola with a vertical axis of symmetry, this equation typically takes the form $y=a{x}^2$ where $a$ is a constant that defines the parabola's curvature.

• In the context of a searchlight, adapting the equation helps to understand light reflection characteristics.
• The parameter $a$ is particularly important, as it determines the spread and steepness of the parabola. Small values of $a$ create wider curves, while larger values create narrow ones.

In this example, the equation helps calculate the depth of the paraboloid, a crucial measure for engineering tasks. Knowing the position and significance of $a$ helps in applications requiring precision, like determining how far a light beam reaches.