A microhpone is palced at the focus of the parabolic reflector to collect sounds for the television broadcast of the World Cup Soccer final game. The focus of the parabola that is a cross section of the refector is 6 inches from the vertex. The latus rectum is 24 inches in length. Assume the focus is at the origin and the parabola opens to the right. Write an equation for the cross section.

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The vertex equation for a parabola is:

`y=a(x-h)^2+k` , where (h,k) is the vertex

Focus is at the origin (0,0)

Since the parabola opens to the right we know that the vertex is 6 inches to the left of the focus. Therefore the vertex is at (-6,0)

In order to determine the value for a we can use the knowledge that the length of the focus is equal to 1/a:

`1/a=24 -> a =1/24`

Therefore, the equation for this parabola is:

`y=1/24(x-(-6))^2+0`

`y=1/24(x+6)^2`

Expanding, this gives us the above equation in standard form:

`y=1/24(x^2+12x+36)`

`y=1/24x^2+1/2x+3/2`

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