Question

A satellite dish has the shape of a parabola when viewed from the side. The receiver (which is at the focus) is 8 centimeters from the bottom of the dish and the dish is 98 centimeters deep. How wide is the dish?

Answer 1

The width of the satellite dish, which has a parabolic shape, is 56 centimeters. This measurement is derived using the properties of a parabola and the given distance of the receiver (focus) from the vertex.

Step-by-step explanation:

The shape of the satellite dish can be described using the properties of a parabola. The focus of a parabola is located at a point that is at an equal distance from any point on the parabola to its directrix (a line perpendicular to the axis of symmetry of the parabola). In this case, the focus is 8 centimeters from the vertex of the parabola because it is said to be 8 centimeters from the bottom of the dish. The depth of the dish is 98 centimeters, which is the distance from the vertex to the directrix. Therefore, the focal length (distance from the vertex to the focus) is 98 cm / 2 = 49 cm (since the focus is halfway between the vertex and the directrix).

Using the focal length f, the width w of the dish at its opening can be found with the parabolic equation x2 = 4fy, where y is the depth and x is half of the width of the dish. So w/22 = 4 * 49 * 98, therefore, w/2 = √(4 * 49 * 98), and w = 2 * √(4 * 49 * 98).

By calculating, we get w = 2 * 28, which equals 56 centimeters. So the width of the satellite dish is 56 centimeters.