Question

An engineer designs a satellite dish with a parabolic cross section.

The dish is 10 ft wide at the opening, and the focus is placed 5 ft from
the vertex.
a) Position a coordinate system with the origin at the vertex and
the x-axis on the parabola's axis of symmetry and find an equation of
the parabola.
b) Find the depth of the satellite dish at the vertex.

Answer 1

The equation of the parabola for the satellite dish is y = 1/20 x^2, with the coordinate system origin at the vertex, and the depth of the satellite dish at the vertex is 1.25 ft.

Step-by-step explanation:

To find the equation of the parabola for a satellite dish with a parabolic cross section where the width is 10 ft across and the focus is 5 ft from the vertex, we position the coordinate system with the origin at the vertex of the parabola. The parabola's axis of symmetry coincides with the x-axis so that the focus is at (0, 5). From the properties of a parabola, we know the distance from the focus to the directrix is the same as from the focus to the vertex which is 5 ft. Hence, the directrix is at y = -5.

The standard form of a parabolic equation is y = ax^2. Considering that the parabola opens upward and the focal distance f is 5 ft, we have 1/(4f) = a or a = 1/20. Therefore, the equation of the parabola becomes y = 1/20 x^2.

The dish's depth at the vertex can be seen as the y-coordinate of the intersection of the parabola and the line x = 5 because the parabola is 10 ft wide and symmetrical about the y-axis. Substituting x = 5 into the parabolic equation, we get y = 1/20(5)^2 = 1/20(25) = 1.25 ft. Therefore, the depth of the satellite dish at the vertex is 1.25 ft.