Question

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 parabola is 24 inches wide and 4 inches deep, how far from the vertex should the microphone be placed? (1 point)

4 inches

18 inches

9 inches

12 inches

Answer 1

Answer: 9 inches.

Explanation:

Given: The width of the parabola= 24 inches

The depth of the parabola = 4 inches

Assume the parabola open upwards and the vertex of the parabola be (0,0)

Then the point on parabola (x,y)=
`(-(24)/(2),4)\ and (+(24)/(2),4)`

⇒ (-12,4) and (12,4) are points on parabola.

Equation of parabola opens upwards with vertex=(0,0) is
`x^2=4ay`

Put (12,4) in the equation, we get

`(12)^2=4a(4)\\\Rightarrow\ 144=16a\\\Rightarrow\ a=(144)/(16)=9\\\Rightarrow\ a=9`

Since, the microphone itself is placed on the focus of the parabola.

Hence, the microphone is 9 inches far from the vertex.

Answer 2

we are given a microphone that has a shape of a parabola in which the dimensions are 24 inches wide and 4 inches deep. In this case, we can put the vertex at the origin, that is Place the vertex at the origin. Then the parabola has equation
4ay = x²
4a(4) = 12^2
a = 9 in
THe answer then is C.