Question 1186192
Here's how to find the location of the receiver and the equation of the paraboloid:

**1. Set up a coordinate system:**

It's easiest to place the vertex of the paraboloid at the origin (0,0) and have the parabola open upwards.  This makes the axis of symmetry along the y-axis.

**2. Identify key points:**

*   The dish is 12 feet across, so it extends 6 feet to either side of the y-axis.  This gives us two points on the parabola: (-6, 3) and (6, 3).
*   The dish is 3 feet deep, so the vertex is at (0,0) and the focus is somewhere along the positive y-axis.

**3. Standard equation of a parabola:**

The standard form equation of a parabola opening upwards with its vertex at the origin is:

x² = 4py

where 'p' is the distance from the vertex to the focus.

**4. Solve for 'p' (the distance to the focus):**

We can use one of the points we identified, such as (6, 3), and plug it into the equation to solve for 'p':

6² = 4p * 3
36 = 12p
p = 3

**5. Location of the receiver (focus):**

Since the vertex is at (0,0) and p = 3, the focus is located at (0, 3).  This means the receiver should be placed 3 feet above the vertex of the dish.

**6. Standard form equation of the paraboloid:**

Now that we know p = 3, we can plug it into the standard equation:

x² = 4 * 3 * y
x² = 12y

Therefore, the location of the receiver (focus) is **3 feet above the vertex**, and the standard form equation of the paraboloid is **x² = 12y**.