Question

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 8 inches from the vertex. Determine the width of the dish if the depth is to be 3 feet. (Round your answer to two decimal places.)

Answer 1

w≅8.46ft

Explanation:

Data

focus=8

vertex=(0,0)

width=?

depth (y)=3ft

parabola equation:
`(x-h)^(2)=4p(y-k)`; then for V=(0,0),
`x^(2) =4py`, be focus(f)=8in;
`x^(2) =4*8*y=32y` →
`y=(x^(2) )/(32)`

we have that y=3ft then
`3=(x^(2) )/(32)`; but 1in≅0.083ft so
`x^(2) =3*32*0.083=7.968` → x=±2.82, finally w=3*2.82≅8.46ft