Parabola problem help?
The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 centimeters and its depth is 1 centimeter. How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?
Comments
Diameter = 4 cm.
Depth = 1 cm.
The diameter of the parabola defines a point P(x, y) as:
P(4, 0):
x = 4
y = 0
Half the diameter defines the x-value of the vertex, h, as
h = 1/2 x
h = 1/2 (4)
h = 2
The depth, k, defines the y-value, k, of the vertex:
k = 1
Equation of Parabola:
y = a (x - h)² + k:
0 = a (4 - 2)² + 1
0 = a (2)² + 1
0 = a (4) + 1
0 = 4a + 1
4a = - 1
a = - 1/4
p = 1 / 4a:
p = 1 / 4(- 1/4)
p = 1 / - 1
p = - 1
The light bulb should be placed 1 cm. from the vertex.