A parabolic headlight is formed by revolving the parabola y^2 = 15x between the lines y = -3 and y = 3 about its line of symmetry. Where should the headlight bulb be placed for maximum illumination?
Maximum illumination will be achieved if the bulb is placed at the focus of the parabola created by the cross-section of the headlight.
If we rearrange the equation we know that:
`x=1/15y^2` , therefore, a = 1/15
The focus of a parabola is located at:
If we place the vertex of the parabola at the origin (0,0) for simplicity this gives:
Substituting our known value of a we get:
Therefore, the bulb should be placed 3.75 units to the right of the vertex.