Problem 62 - College Algebra

Question

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

Step-by-Step Solution

Answer

The light bulb should be placed \( \frac{1}{16} \) inches from the vertex of the flashlight reflector.

Step 1: Understand the Parabola

The flashlight's reflector is in the form of a parabola. So, it can be represented by the equation \(y = a \cdot x^2\), where 'x' is the distance from the vertex to the casting edge and 'y' is the depth of the parabolic shape. The light bulb should be placed at the focus of the parabolic reflector, which is at a distance 'a' along the 'y' axis.

Step 2: Substitute the known values

Here, the diameter of the casting is given as 8 inches. Therefore the value of 'x' will be half of it, that is 4 inches. The depth 'y' is given as 1 inch. Substitute these into the equation:

Step 3: Solve for 'a'

By substituting x = 4 and y = 1 into the formula, we get: \(1 = a \cdot 4^2\). By simplifying the equation, the value of 'a' will be \( \frac{1}{16} \) inches.

Step 4: Interpreting the Solution

The calculated value 'a' represents the distance from the vertex where the bulb should be placed.