We know that a microphone is placed at the focus of a parabolic reflector. Also, we know that the mic is placed at the origin and it is 6 inches away from the vertex of the cross section. The cross section of a parabolic reflector is a parabola that opens to the right, and we need to find its equation. Let's make a graph to visualize the situation.
Let's recall the standard form of the equation of a parabola with a horizontal axis of symmetry.
x=a(y-k)^2+h
In this form, the focus of a parabola is the point (h+ 14a,k). At the same time, we know that the focus of the parabola is the origin, (0,0). We can combine these pieces of information to make an equation.
(h+ 14a,k)=(0,0)
⇕
h+ 14a=0 & (I) k=0 & (II)
We found that k=0. Let's solve Equation (I) to find the relationship between h and a.
h+ 14a=0 k=0
⇕
h=- 14a k=0
Now, we can substitute the values k= 0 and h= - 14a into the equation of the parabola.
x=a(y-k)^2+h
⇕
x=a(y- 0)^2+( -1/4a)
To find the equation of the cross section, we need to find the value of a. Let's look at the graph once again.
The coordinates of the vertex of a parabola in standard form are (h,k). From the graph, we can see that the first coordinate of the vertex is -6, therefore h=-6. Also, earlier we found that h=- 14a. We can combine this information and create an equation for a.
h=-6 h=- 14a ⇒ -6=- 14a
Let's solve this equation!
We found that a= 124, h= -6, and that k= 0. We can substitute these values into the standard form of the parabola equation to find the equation of the cross section.
x=a(y-h)^2+k
⇕
x= 1/24(y-( 0))^2+( -6)
⇕
x=1/24y^2-6
