Chapter 10, Problem 39RE

To determine

To find: How deep should the mirror be in order to reflect the light rays parallel to the axis of symmetry.

Answer to Problem 39RE

Depth of the mirror should be $0.25$ feet or 3 inches.

Explanation of Solution

Given:

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.

Calculation:

Let $\left(0,0\right)$ be the vertex of the parabola and let it open up.

The equation of the parabola is $x^2=4ay$.

The light source is located 1 foot from the vertex along the axis of symmetry.

Therefore, $a=1$.

Hence, the equation is $x^2=4y$.

The width of opening is 2 so that $\left(1, y\right)$ is located on the parabola.

Substituting $\left(1, y\right)$ in $x^2=4y$, we get,

$1^2=4y$, $y=\frac{1}{4}$

$y=0.25$ feet

The mirror is $0.25$ feet, or 3 inches.