SOLUTION: NASA is designing a new satellite to go on the international space station. The satellite disk is the shape of a parabola. The satellite will be attached to the station on a pole a
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Question 1171303: NASA is designing a new satellite to go on the international space station. The satellite disk is the shape of a parabola. The satellite will be attached to the station on a pole and will place the vertex of the satellite 4 feet away from the surface of the station. The receiver is to be positioned 7 feet above the roof. Write an equation that best models the parabolic cross section of the satellite dish.( can I see how you solve it please?)
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Absolutely! Let's break down how to find the equation for the parabolic cross-section of the satellite dish.
**1. Set up a Coordinate System**
To make this problem easier, let's set up a coordinate system:
* Place the vertex of the parabola at the origin (0, 0).
* Let the axis of symmetry of the parabola be the y-axis.
* Since the receiver is above the vertex, the parabola opens upwards.
**2. Understand the Given Information**
* The vertex is 4 feet away from the surface of the station, and the receiver is 7 feet above the vertex.
* This means the receiver is located at the point (0, 7).
* The standard equation of a parabola that opens upwards with its vertex at the origin is:
* x² = 4py
* Where 'p' is the distance from the vertex to the focus (the location of the receiver).
**3. Find the Value of 'p'**
* In this case, the distance from the vertex (0, 0) to the receiver (focus) (0, 7) is 7 feet.
* Therefore, p = 7.
**4. Write the Equation**
* Substitute p = 7 into the standard equation:
* x² = 4(7)y
* x² = 28y
**Therefore, the equation that best models the parabolic cross-section of the satellite dish is x² = 28y.**
**Visual Explanation**
Imagine the satellite dish in a 2D plane:
* The vertex is at (0, 0).
* The receiver (focus) is at (0, 7).
* The parabola opens upwards along the y-axis.
The equation x² = 28y represents this parabolic shape.
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