Question

You know that functions can be used to model any number of phenomena and systems. We have seen that transformations allow functions to shift around on the coordinate plane, to stretch, compress, and flip. Choose one of the projects below and study how you can transform functions to meet real-world challenges.

Radio Telescope:
Because of its special reflective properties, a parabola is an ideal shape for collecting weak radio signals. The European Space Agency has asked you to write an equation for the parabolic dish of their new radio telescope.It is parallel to the ground and should have the following dimensions:

Width of the dish: 60 m
Depth of the dish: 10 m
Height above ground level measured from the bottom of the dish: 100 m
Horizontal distance from the bottom of the dish to the control room (the origin): 100 m

Answer 1

To write the equation for the parabolic dish of the radio telescope, use the vertex form of a parabola equation and substitute the given dimensions and positions. The resulting equation is y = (1 / 50)x^2.

Step-by-step explanation:

To write an equation for the parabolic dish of the radio telescope, we need to understand how a parabola can be defined using its dimensions. In this case, the width of the dish is 60 m, the depth of the dish is 10 m, and the height above ground level is 100 m measured from the bottom of the dish. The horizontal distance from the bottom of the dish to the control room is 100 m.
Since the dish is parallel to the ground, we can assume that the vertex of the parabola is at the origin (0, 0). The opening of the parabola will be upward since the dish is concave.
Using the vertex form of a parabola equation, y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates, we can substitute the values to find the equation that models the parabolic dish of the radio telescope.

Here's how we can do it:

1. The vertex coordinates (h, k) is (0, 0) since the dish is parallel to the ground.
2. The focus is located at (0, p), where p is the distance from the vertex to the focus. In this case, p = (Width - Depth) / 4 = (60 - 10) / 4 = 12.5 m.
3. By using the focal length, f = p, we can find the value of a, which is the coefficient that determines the width and shape of the parabola. The formula is a = 1 / (4f).
4. Substituting the values for the equation, we get y = (1 / (4 * 12.5))(x - 0)^2 + 0, which simplifies to y = (1 / 50)x^2.

Therefore, the equation for the parabolic dish of the radio telescope is y = (1 / 50)x^2.