Question

a spotlight is made by placing a strong light bulb inside a reflective paraboloid formed by rotating the parabola ( x^{2}=7y ) around its axis of symmetry (assume that ( x ) and ( y ) are in units of inches). in order to have the brightest, most concentrated light beam, how far from the vertex should the bulb be placed? express your answer as a fraction or a decimal rounded to two decimal places.

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens upwards is \(x^{2}=4py\).

Step2: Compare with the given equation

Given \(x^{2}=7y\), comparing with \(x^{2}=4py\), we have \(4p = 7\).

Step3: Solve for \(p\)

Divide both sides of \(4p=7\) by \(4\), so \(p=\frac{7}{4}=1.75\).

Answer:

\(1.75\) inches