43. The antenna of a parabolic dish is 20 meters above its vertex. What is an equation to dish with vertex (0,0)?

Step 1 ：The parabolic dish is a parabola that opens downwards. The standard form of a parabola that opens downwards is given by the equation \(y = -ax^2 + k\), where a is a positive constant, and k is the y-coordinate of the vertex. In this case, the vertex is at the origin (0,0), so k = 0. The antenna is 20 meters above the vertex, so when x = 0, y = 20. This gives us the equation \(y = -ax^2 + 20\).

Step 2 ：To find the value of a, we can set x = 1 (since the parabola is symmetric about the y-axis), and solve for a. The solution for a is 0, which means the equation of the parabola is \(y = -0*x^2 + 20\), or simply \(y = 20\).

Step 3 ：This means the parabolic dish is a horizontal line 20 meters above the vertex. This seems strange, as we would expect a parabolic shape, not a straight line. However, this is because we only have information about the height of the antenna above the vertex, and no information about the width or depth of the dish. Therefore, based on the given information, the equation \(y = 20\) is the best we can do.

Step 4 ：Final Answer: The equation of the dish with vertex (0,0) is \(\boxed{y = 20}\).