Problem 61
Question
A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.
Step-by-Step Solution
Verified Answer
The dish is 43.82 inches wide at a depth of 2 feet.
1Step 1: Understanding the Problem
The problem describes a parabolic dish. The focus is located 5 inches from the vertex. We want to find the width of the dish at a depth (height) of 2 feet (24 inches).
2Step 2: Equation of a Parabola
The standard form of the equation of a parabola with vertex at the origin and focus \((0, p)\) is \(x^2 = 4py\). In this case, \(p = 5\) because the focus is 5 inches above the vertex.
3Step 3: Substituting the Known Values
Substitute \(p = 5\) into the equation, \(x^2 = 4(5)y\), which simplifies to \(x^2 = 20y\).
4Step 4: Determine the Width at 24 Inches Depth
Set \(y = 24\) because we want to find the width at this depth. Substitute \(y = 24\) into the equation so that \(x^2 = 20(24)\).
5Step 5: Calculate the Width of the Dish
Solve \(x^2 = 480\) for \(x\). \( x = \sqrt{480} \approx 21.91 \) inches. Thus, the complete width of the dish is \(2x = 2(21.91) = 43.82\) inches.
Key Concepts
Parabola EquationFocus of ParabolaWidth of ParabolaVertex of Parabola
Parabola Equation
The parabola equation is a crucial concept when working with parabolic shapes, such as the sound receiving dish mentioned in the problem. A parabola is a symmetric plane curve, and the standard equation for a parabola with a vertex at the origin is given by \(x^2 = 4py\). Here, \(p\) represents the distance from the vertex to the focus of the parabola. This form of the equation is useful because it can be adapted based on the location of the vertex and focus.
When analyzing problems involving parabolas, determining the equation allows you to predict and calculate other features, like the width at a certain depth. It's important to understand that the parameters in the equation \(x^2 = 4py\) directly communicate the parabola's geometry. This specific format suits a parabola opening upwards, ideal for describing activities such as understanding the curvature of a dish used in acoustics or satellite technology.
When analyzing problems involving parabolas, determining the equation allows you to predict and calculate other features, like the width at a certain depth. It's important to understand that the parameters in the equation \(x^2 = 4py\) directly communicate the parabola's geometry. This specific format suits a parabola opening upwards, ideal for describing activities such as understanding the curvature of a dish used in acoustics or satellite technology.
Focus of Parabola
The focus of a parabola is a fixed point used in the definition of the parabola itself. The significance of the focus in a sound receiving dish, or any parabolic mirror, cannot be overstated. Any parallel sound waves entering the dish will reflect and pass through the focus.
In our problem, the focus is located 5 inches away from the vertex.
Here’s a trick to remember: the focus is always located "inside" the parabola. In our given problem, understanding the position of the focus relative to the vertex, allows us to determine that \(p = 5\) in the parabola's equation \(x^2 = 4py\), thus focusing sound waves efficiently.
In our problem, the focus is located 5 inches away from the vertex.
Here’s a trick to remember: the focus is always located "inside" the parabola. In our given problem, understanding the position of the focus relative to the vertex, allows us to determine that \(p = 5\) in the parabola's equation \(x^2 = 4py\), thus focusing sound waves efficiently.
- Used to focus incoming parallel sound waves.
- Determines "p" in the parabola equation.
- Located inside the curve for maximum efficiency.
Width of Parabola
The width of a parabola refers to the distance across the parabola at a certain depth or height. In the context of the sound receiving dish problem, we are looking for the dish's width at a height (or depth) of 24 inches. This requires solving the parabola equation for \(x \), which gives the horizontal distance of the parabola. Once solved, you double \(x\) to find the full width.
For example, when we set the depth \(y = 24\), and rearrange \(x^2 = 20 \times 24 \) (using the parabola equation), we'll solve for \(x\) to get the half-width. Finally, multiplying \(x\) by 2 provides us the full width of the dish. In such problems, always remember to convert units consistently, as mixing feet with inches can cause errors.
For example, when we set the depth \(y = 24\), and rearrange \(x^2 = 20 \times 24 \) (using the parabola equation), we'll solve for \(x\) to get the half-width. Finally, multiplying \(x\) by 2 provides us the full width of the dish. In such problems, always remember to convert units consistently, as mixing feet with inches can cause errors.
Vertex of Parabola
The vertex is the point on the parabola that is either the highest or the lowest, depending on its orientation. It is a key reference point in the definition and graphing of a parabola equation. In this situation, the vertex is at the origin \((0, 0)\), which simplifies calculations.
The vertex serves as the starting point to measure distances, like the focus's position or any specific point along the parabola. In terms of the problem, the vertex is 5 inches below the focus, providing symmetry. This helps visualize and solve the problem more effectively, as knowing precisely where the vertex is located simplifies identifying important parabola characteristics like depth and width.
The vertex serves as the starting point to measure distances, like the focus's position or any specific point along the parabola. In terms of the problem, the vertex is 5 inches below the focus, providing symmetry. This helps visualize and solve the problem more effectively, as knowing precisely where the vertex is located simplifies identifying important parabola characteristics like depth and width.
- Indicates the parabola's turning point.
- Essential in defining the parabola's equation.
- Helps establish the orientation and dimension references.
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