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Question 1198216: 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 72 feet across at its opening and 6 feet deep at its center, where should the receiver be placed? (Hint: Draw a cross section of the dish on a graph and place the vertex at (0, -6) so that the opening of the dish lies on the x-axis).
Find the equation of the parabola.
How far above the vertex should the receiver be placed?
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! **1. Find the Equation of the Parabola**
* **Vertex:** (0, -6)
* **Point on the Parabola:** (36, 0) (Since the dish is 72 feet across, the point on the parabola is half of that, or 36 feet, from the vertex)
* **Standard Form of a Parabola:**
(x - h)² = 4p(y - k)
where (h, k) is the vertex
* **Substitute values:**
(x - 0)² = 4p(y - (-6))
x² = 4p(y + 6)
* **Find the value of 'p':**
Substitute the point (36, 0) into the equation:
36² = 4p(0 + 6)
1296 = 24p
p = 1296 / 24
p = 54
* **Equation of the Parabola:**
x² = 4 * 54 * (y + 6)
x² = 216(y + 6)
**2. Find the Distance of the Receiver from the Vertex**
* The receiver should be placed at the focus of the parabola.
* The distance from the vertex to the focus is 'p'.
* **Receiver Distance:** 54 feet
**Therefore:**
* The equation of the parabola is: x² = 216(y + 6)
* The receiver should be placed 54 feet above the vertex of the dish.
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