Chapter 10.2, Problem 65AYU

Constructing a Flashlight The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 inches and its depth is 1 inch. How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?

To determine

To find: How far from the vertex should the light bulb be placed, if the diameter is 4 inches and its depth is 1 inch, so that the rays will be reflected parallel to the axis?

Answer to Problem 65AYU

Light bulb should be placed 1 inch from the vertex.

Explanation of Solution

Given:

The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 inches and its depth is 1 inch.

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 $\text{a}$ is the distance from vertex to focus.

The parabola is 4 feet across at its opening and 1 feet deep.

The points $\left(2,1\right)$ and $\left(-2,1\right)$ are the points on the parabola.

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

$2²=4*a*1$

$a=1=1$ inch.

The light bulb should be placed 1 inch from the vertex so that the rays will be reflected parallel to the axis.