Question

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 56 feet across at its opening and 7 feet deep at its center, where should the receiver be placed?
find the equation of the parabola.
how far above the vertex should the receiver be placed?
select an answer
question help: video written example

Explanation:

Step1: Define the parabola equation

We place the vertex of the parabola at the origin \((0,0)\) and open upwards, so the standard form of the parabola is \(x^{2}=4py\), where \(p\) is the distance from the vertex to the focus (the location of the receiver).

Step2: Determine the point on the parabola

The dish is 56 feet across at the opening and 7 feet deep at the center. So when \(y = 7\), \(x=\frac{56}{2}=28\) (since it's 56 feet across, the x - coordinate at the edge of the opening is half of 56). The point \((28,7)\) lies on the parabola.

Step3: Substitute the point into the equation

Substitute \(x = 28\) and \(y=7\) into the equation \(x^{2}=4py\):

$$
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$$

Step4: Solve for \(p\)

Divide both sides of the equation \(784 = 28p\) by 28:
\(p=\frac{784}{28}=28\)

For the equation of the parabola, substituting \(p = 28\) into \(x^{2}=4py\), we get \(x^{2}=112y\) (since \(4p=4\times28 = 112\)).

Answer:

The equation of the parabola is \(x^{2}=112y\), and the receiver should be placed 28 feet above the vertex.