Question

a lamp with a parabolic reflector is shown in the figure. the bulb is placed at the focus and the focal diameter is 12 cm. (assume the vertex o is at the origin. let the x - axis be horizontal and the y - axis be vertical.)
(a) find the standard equation of the parabola.
(b) find the diameter of c, d) of the opening (in cm), 22 cm from the vertex. (round your answer to the nearest whole number.)

Explanation:

Step1: Determine the standard form of the parabola

Since the parabola opens downward and the vertex is at the origin \((0,0)\), the standard form of the parabola is \(x^{2}=-4py\).
We know that the focal diameter is \(12\) cm. For the parabola \(x^{2}=-4py\), the length of the focal diameter is \(4p\). So, \(4p = 12\), which gives \(p = 3\).

Step2: Write the standard equation of the parabola

Substitute \(p = 3\) into the standard form \(x^{2}=-4py\).
We get \(x^{2}=-12y\).

Step3: Find the diameter \(d(C,D)\)

We want to find the \(x\) - values when \(y=-22\).
Substitute \(y = - 22\) into the equation \(x^{2}=-12y\).
We have \(x^{2}=-12\times(-22)=264\).
Then \(x=\pm\sqrt{264}\approx\pm16.25\).
The diameter \(d(C,D)=2|x|\).
So \(d(C,D)=2\sqrt{264}\approx32.5\approx33\)

Answer:

(a) \(x^{2}=-12y\)
(b) \(33\)