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We are asked to find the location of the receiver in a parabolic dish that is 6 feet wide at its opening and 2 feet deep. Since the dish is a paraboloid of revolution, the cross-section of the dish can be modeled by a parabola in the form \( y = ax^2 \), where the vertex of the parabola is at the origin, and the receiver is placed at the focus of this parabola.

### Step-by-Step Solution:

1. **Setting up the equation**:
   The parabola has the vertex at the origin, and the dish is 6 feet wide across the top, meaning the distance from the vertex to the edge is 3 feet (half of 6 feet). The dish is also 2 feet deep, meaning the point on the parabola at \( x = 3 \) corresponds to \( y = -2 \) (since the parabola is "upside down").

   Therefore, we know that at \( x = 3 \), \( y = -2 \), so:
   \[
   -2 = a(3)^2 \quad \text{or} \quad -2 = 9a.
   \]
   Solving for \( a \), we get:
   \[
   a = -\frac{2}{9}.
   \]

2. **Equation of the parabola**:
   The equation of the parabola is:
   \[
   y = -\frac{2}{9}x^2.
   \]

3. **Finding the focus**:
   The standard form of a parabola is \( y = \frac{1}{4p}x^2 \), where \( p \) is the distance from the vertex to the focus. Comparing this with the equation \( y = -\frac{2}{9}x^2 \), we have:
   \[
   \frac{1}{4p} = -\frac{2}{9}.
   \]
   Solving for \( p \), we get:
   \[
   4p = -\frac{9}{2} \quad \Rightarrow \quad p = -\frac{9}{8}.
   \]
   Thus, the focus is located \( \frac{9}{8} \) feet, or 1.125 feet, above the vertex (since the negative sign reflects the downward direction of the parabola).

### Conclusion:
The receiver, placed at the focus, is located **1.125 feet above the vertex** of the dish.

Would you like more details on this solution or have any other questions?

Here are 5 related questions:
1. How do parabolas model the shape of satellite dishes in real life?
2. What are the properties of a paraboloid of revolution?
3. How is the depth of the dish related to the focal length of the parabola?
4. Can this method be applied to other cross-sectional shapes of dishes, like ellipses?
5. How would changing the width or depth of the dish affect the location of the focus?

**Tip**: The focus of a parabolic dish is crucial for reflecting signals to a single point, making it ideal for satellite dishes and telescopes.

A cable TV receiving dish is in the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 6 feet across at its opening and 2 feet deep.

Learn how to find the location of a receiver in a parabolic dish that is 6 feet wide and 2 feet deep. This problem explains the geometry of parabolas, focus, and vertex relationships, suitable for high school students.

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