Problem 70 - Algebra and Trigonometry

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 1/16 inches from the vertex.

Step 1: Define the parabola

A parabola is defined as the set of all points that are an equal distance from the focus and a line called the directrix. The vertex is halfway between the focus and the directrix. Given a parabola that opens upward and with vertex at origin, the equation of the parabola is given by \(y =4af\), where 'a' is the distance from the vertex to the focus and 'f' is any point on the x-axis or the 'x' value.

Step 2: Compute diameter and depth

The diameter of the reflector is 8 inches. Therefore, the width or 'x' value of the parabola is 4 inches (half of the diameter). The depth or 'y' value of the parabola is 1 inch. These values replace the 'f' (x-value) and 'y' values in the equation. Therefore, \( 1 = 4a*4 \)

Step 3: Solve for 'a

Solving the equation for 'a', we find that \( a= \frac{1}{16} \) inches. This value of 'a' is the distance from the vertex of the parabola to the focus. Thus, this is how far the lightbulb should be placed from the vertex.