a) Well, well, looks like we have a math question here. Let's dive right into it. Now, since we have the opening width and the focus of the parabolic dish, we can get cracking on that equation.
In our coordinate system, let's set the vertex as the origin. Now, since the x-axis is on the parabola's axis of symmetry, we can simplify our equation. The focus typically sits on the axis of symmetry of a parabola, so we know the focus will be located at (8,0).
We also know that the width of the dish at the opening is 10 ft. So, if we assume the vertex is (0,0) and the focus is at (8,0), we can take half of the width to get the distance from the vertex to the opening, which is 5 ft.
Now, let's think about the parabolic equation, which takes the form y = ax^2. Since our focus is located at (8,0), we know that every point (x,y) on the parabola should satisfy the distance formula from the vertex to the focus. In this case, it is sqrt((x - 8)^2 + y^2) = 5.
Now it's time to solve for y. We square both sides of our equation like a mad scientist, and we end up with (x - 8)^2 + y^2 = 25. And there you have it, my friend! The equation of the parabola is (x - 8)^2 + y^2 = 25. Let's move on to our next challenge!
b) Ah, the depth of the satellite dish at the vertex, you say? Well, if we refer back to our equation (x - 8)^2 + y^2 = 25, the vertex would be located at (8,0). So the depth of the dish at the vertex would simply be the y-coordinate of the vertex. And guess what, my friend? It's a big fat zero! The dish is flat at the vertex. So the depth of the satellite dish at the vertex is a resounding zero!