Chapter 11, Problem 12T

A parabolic reflector for a car headlight forms a bowl shape that is 6 in. wide at its opening and 3 in. deep, as shown in the figure at the left. How far from the vertex should the filament of the bulb be placed if it is to be located at the focus?

To determine

To find: The distance from the vertex the filament of the bulb be placed if it is to be located at the focus.

Answer to Problem 12T

The filament of the bulb should be placed at a distance of $\frac{3}{4}\text{in}$ from the vertex of the reflector.

Explanation of Solution

Given:

The parabolic reflector for a car headlight forms a bowl shape with 6in. wide at the opening and 3 in deep.

Definition used:

 “The equation of the parabola with vertex $V(0,0)$, focus $F(p,0)$ and directrix $x=-p$ is $y^2=4px$”.

Calculation:

Observe that from the given figure, that the parabola is open rightward, wide at the opening is 6 in and depth is 3 in.

Let the equation of the reflector be $y^2=4px$ with vertex at the origin $V(0,0)$, then the x-axis is the axis of the parabola.

Hence, the coordinate of the point intersects the axis and the diameter is $C(3,0)$ and the end points of the diameter at the opening are $A(3,-3)$ and $B(3,3)$ are as shown in the above Figure.

The points $A(3,-3)$ and $B(3,3)$ are on the parabola, $y^2=4px$, that is the points satisfies equation of the parabola $y^2=4px$.

Substitute $(3,3)$ in the equation of the parabola $y^2=4px$.

$\begin{array}{c}{y}^2=4px\\ 3^2=4p\left(3\right)\\ 9=12p\\ p=\frac{9}{12}\end{array}$

$=\frac{3}{4}$

By the above definition, the focus of the parabola is $F\left(\frac{3}{4},0\right)$.

Since the filament of the reflector should be placed at the focus, the filament is placed at a distance of $\frac{3}{4}\text{in}$ from the vertex.