Question

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 84 feet across at its opening and 7 feet deep at its center, where should the receiver be placed?

Answer 1

Explanation:

Since the satellite dish is shaped like a paraboloid of revolution, we can use the formula for the standard equation of a parabola:y = (1/4p) x^2where p is the distance from the vertex to the focus.Let's first find the equation of the parabola that represents the satellite dish. We know that the vertex is at (0,0) (since the dish is symmetric) and that the opening is 84 feet across, which means that the distance between the two x-intercepts is 84. So the equation of the parabola is:y = (1/168) x^2Now we need to find the value of p. We know that the dish is 7 feet deep at its center, which means that the vertex of the parabola is 7 feet below the opening. Therefore, the distance from the vertex to the focus (which is where the receiver should be placed) is:p = 7 + (84/4) = 28So the receiver should be placed 28 feet from the vertex of the parabola (which is the center of the dish).