Chapter 10.2, Problem 63AYU

Satellite Dish A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and 4 feet deep at its center, at what position should the receiver be placed?

To determine

To find: The distance at which the receiver to be placed, if the dish is 10 feet across at its opening and 4 feet deep at its center.

Answer to Problem 63AYU

$1.5625$ feet from the base of the dish.

Explanation of Solution

Given:

Satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. The dish is 10 feet across at its opening and 4 feet deep at its center.

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(0,0\right)$	$\left(0,a\right)$	$y=-a$	$x²=4ay$	Parabola, axis of symmetry is $y\text{-axis}$, opens up.

Calculation:

Let the vertex of the parabola be $\left(0,0\right)$ and let it open up.

The equation of the parabola is given by $x²=4ay$ where $a$ is the distance from vertex to focus.

The parabola is 10 feet across and 4 feet deep, the points $\left(5,4\right)$ and $\left(-5,4\right)$ are the points on the parabola.

Substituting the points in the equation $x²=4ay$, we get,

$5²=4*a*4$

$a=\frac{25}{16}=1.5625=1.5625$ feet.

The receiver is located at the focus which is $1.5625$ feet from the base of the dish along the axis of symmetry $y\text{-axis}$.