SOLUTION: the reflecting dish of a parabolic microphone has a cross section in the shape of a parabola. The microphone itself is placed on the focus of the parabola. If the parabolas is 40

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: the reflecting dish of a parabolic microphone has a cross section in the shape of a parabola. The microphone itself is placed on the focus of the parabola. If the parabolas is 40       Log On


   

Question 1024755: the reflecting dish of a parabolic microphone has a cross section in the shape of a parabola. The microphone itself is placed on the focus of the parabola.
If the parabolas is 40 inches wide, and 20 inches deep,
How far from the vertex should the microphone be placed
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Put a graph of the parabola shape onto coordinate system with vertex on the origin. Face the parabola concave upward. Two other points will be (-20,20) and (20,20).

Find the standard form equation.
y=a%28x-h%29%5E2%2Bk.
(h,k) will be (0,0), the origin according to planning.
The form is now as y=ax%5E2.
Use either of the other points to determine the coefficient's value.
-
a=y%2Fx%5E2
a=20%2F20%5E2
a=1%2F20
-
The equation in standard form more specifically is y=%281%2F20%29x%5E2.

Change the form of this equation to what you might expect if you had directrix and focus and were to have used Distance Formula to derive an equation.
y%2A20=%281%2F20%29x%5E2%2A20
20y=x%5E2 or x%5E2=20y.
Remember, you are looking for the focus.

The distance p is how far the focus is from the vertex.
The derivation from the definition of a parabola would give highlight%28x%5E2=4py%29; and you will compare this to highlight%28x%5E2=20y%29.

4p=20
p=20%2F4
highlight%28p=5%29

The focus is 5 inches away from the vertex and on the concave side of the dish.


Check your book about this or Equation of parabola using Focus and Directrix, for vertex at the Origin for extra help.