Chapter 10, Problem 77RE

To determine

How deep should the mirror be in order to reflect the light rays parallel to the axis of symmetry?

Answer to Problem 77RE

The light bulb should be placed $\boxed{\frac{1}{4}ft}$ from the vertex along the axis of symmetry, the y-axis.

Explanation of Solution

Given information:

A searchlight is shaped like a paraboloid of revolution. If a light source is located $1$ foot from the vertex along the axis of symmetry and the opening is $2$ feet across, how deep should the mirror be in order to reflect the light rays parallel to the axis of symmetry?

Calculation:

Since, the rays are to be reflected parallel to the axis of symmetry, the light bulb should be placed at the focus of the parabola.

Given that the searchlight is in the shape of paraboloid of revolution with width $2$ inches and depth $1$ inch, consider the vertex to be at the origin and its focus on the positive y-axis.

Since, the width is $2$ inches and depth is $1$ inch, the two points on the parabola are $\left(2,1\right)$ and $\left(-2,1\right)$ .

  

The equation of such a parabola is given by $x^2=4ay$ .

Since, we have the point $\left(1,1\right)$ and the focus as $\left(0,a\right)$ , substitute $1$ for $x$ and $1$ for $y$ in $x^2=4ay$ to find $a$ .

  ${\left(1\right)}^2=4a\left(1\right)$

  $1=4a$

Now, divide both sides by $4$ .

  $\frac{1}{4}=\frac{4a}{4}$

  $a=\frac{1}{4}$

Thus, the focus is at $\left(0,\frac{1}{4}\right)$ . The distance from the vertex to the focus is $a=\frac{1}{4}$ ft or $\frac{1}{4}\left(12\right)=3$ inches.

Hence, the light bulb should be placed $\boxed{\frac{1}{4}ft}$ from the vertex along the axis of symmetry, the y-axis.