Chapter 9.1, Problem 99E

(a)

To determine

To find:The equation for the cross section of the reflector.

Answer to Problem 99E

The equation for the cross section of the reflector is $y^2=6x$ .

Explanation of Solution

Given information:

The filament of an automobile headlight is at the focus of a parabolic reflector, which sends light out in a straight beam.(See figure). The filament of the headlight is $1.5$ inches from the vertex.

  

Calculation:

As clear from the figure that the cross section is in form of a rightwards parabola.

As the filament is placed at focus of the parabola and it is $1.5$ inches from the vertex.

So, the value of $p$ is 1 $1.5$ .

The general equation of the parabola is 3 $y^2=4px$ .

Substitute $1.5$ for $p$ in the equation.

  $\begin{array}{c}{y}^2=4px\\ y^2=4\left(1.5\right)x\\ y^2=6px\end{array}$

Therefore, the equation for the cross section of the reflector is $y^2=6x$ .

(b)

To determine

To calculate:The depth of the reflector.

Answer to Problem 99E

The depth of the reflector is $2.67$ inches.

Explanation of Solution

Given information:

The filament of an automobile headlight is at the focus of a parabolic reflector, which sends light out in a straight beam.(See figure). The reflector is $8$ inches wide.

  

Calculation:

As calculated in part(a) the equation of the cross section is $y^2=6x$ .

As the depth of the reflector is $8\text{in}\text{.}$ , therefore the value of $y=4$ .

Substitute $4$ for $y$ in the equation.

  $\begin{array}{c}{y}^2=6x\\ 4^2=6x\\ x=\frac{16}{6}\\ x\approx 2.67\end{array}$

Therefore, the depth of the reflector is $2.67$ inches.