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A radio telescope has a parabolic dish. Radio signals are collected at the focal point (focus) of the parabola. The distance from the vertex of the parabolic dish to the focus is 20 feet. The vertex of the dish is located at a point 30 feet above the ground and 80 feet to the east of a computer that reads and records data from the telescope. The diameter of the dish is 120 feet. What is the depth of the parabolic dish?

I've calculated: the vertex is (80, 20), the horizontal endpoint of the dish is 140, the vertical endpoint of the dish is 75. But need help to find the depth? Please help!?

Jan 7, 2021

#1
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We can let the parabola open upward......so we will  have the form

4a ( y - k)  =  (x - h)^2

The distance from the vertex to the focus =  20   = a

We can let the vertex be   ( 80, 30)

So we have the form

4(20) ( y- 30)  = (x - 80)^2   simplify

80 ( y - 30)  = ( x - 80)^2

Since the diameter  is 120 ft...the radius is 60 ft.....so we can let one point on the parabola be   ( 80 + 60 , a) =

(140 , a)......where a  is the height of the dish....so we have that

80 ( a - 30) = (140 - 80)^2

80 ( a - 30)  = (60)^2

80 ( a - 30) = 3600      divide both sides by  80

a -  30 =  45      add 30 to both sides

a = 75 ft  = the height of the dish

Jan 7, 2021
#2
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Thank you, that's what I got. How do you calculate for depth?

SMagic  Jan 7, 2021
#3
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Subtract the vertex height (30 ft) from the vertical endpoint (75 ft )    to get the depth    (if you calculated the vertical endpoint correctly)

Jan 7, 2021
#4
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I will just put the bowl shaped parabola's vertex  on the origin   0,0

then parabola standad form becomes      (x-h)^2 = 4p (y-k)      p is given as 20 in this problem

x^2 = 80 y

y = 1/80  x2

Now....at the edge of the disk   x = 60 ( as the diameter is 120 ft)

y = depth = 1/80 (60)2 = 45 ft deep

Jan 7, 2021