Question

Constructing a Headlight - A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight at its opening?

Short answer

The diameter of the headlight at its opening is $4\sqrt{2}$.

The graph of the headlight shown below .

Step-by-step solution

Step 1. Given information .

Consider the given distance and depth.

Step 2.  Equation of parabola used .

The equation of parabola is $x^2=4ay$the vertex of parabola is at origin and the focus is on the y positive axis .

Here $a=1$and $y=2$.

Substitute the values in equation of parabola .

$$\begin{gathered} x^2=4ay \\ {x}^2=4·1·2 \\ {x}^2=8 \\ x=2\sqrt{2} \end{gathered}$$

Therefore the diameter of headlight is$2·2\sqrt{2}=4\sqrt{2}$.

Step 3.  Plot the graph .

The graph of the headlight in the shape of parabola is shown below .