Question

gabe is designing a flashlight that uses a parabolic reflecting mirror and a light source. the shape of the mirror can be modeled by a parabola that opens upward with a diameter of 4 in. and a depth of 1 in. the vertex of the parabola is at the origin. where should gabe place the bulb to ensure a perfect beam of light?
(0,1)
(0,4)
(1,0)
(4,0)

Explanation:

Step1: Write the standard form of parabola

The standard form of a parabola that opens upward with vertex at the origin \((0,0)\) is \(x^{2}=4py\).

Step2: Substitute the given point

Since the diameter is \(4\) in (so \(x = 2\) or \(x=- 2\)) and depth \(y = 1\) in. Substitute \(x = 2\) and \(y = 1\) into \(x^{2}=4py\). We get \(2^{2}=4p\times1\).

Step3: Solve for \(p\)

From \(4 = 4p\), we can solve for \(p\) by dividing both sides by \(4\). So \(p = 1\).

Step4: Recall the property of parabola

The light - bulb (focal point) should be placed at \((0,p)\). Since \(p = 1\), the bulb should be placed at \((0,1)\).

Answer:

\((0,1)\)