Question

the reflector of a flashlight is in the shape of a parabolic surface. the casting has a diameter of 12 inches and a depth of 6 inches. how far from the vertex should the light bulb be placed?
the light bulb should be placed □ inches from the vertex. (simplify your answer.)

Explanation:

Step1: Define the parabola equation

We place the vertex of the parabola at the origin \((0,0)\) and open upwards, so the equation of the parabola is \(x^{2}=4py\), where \(p\) is the distance from the vertex to the focus (where the light bulb should be placed).

Step2: Determine the point on the parabola

The diameter is 12 inches, so the radius is 6 inches, and the depth is 6 inches. So the point \((6,6)\) lies on the parabola (since at a depth of 6 inches (y = 6), the x - coordinate is half of the diameter, which is 6).

Step3: Substitute the point into the equation

Substitute \(x = 6\) and \(y=6\) into the equation \(x^{2}=4py\). We get \(6^{2}=4p\times6\).

Step4: Solve for \(p\)

Simplify the left - hand side: \(36 = 24p\). Then, solve for \(p\) by dividing both sides by 24: \(p=\frac{36}{24}=\frac{3}{2}=1.5\).

Answer:

\(1.5\)