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# Help with 2 questions...

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1) LORAN is a long range hyperbolic navigation system. Suppose two LORAN transmitters are located at the coordinates (-100,0) and (100,0) where unit distance on the coordinate plane is measured in miles  A receiver is located somewhere in the first quadrant. The receiver computes that the difference in the distances from the receiver to these transmitters is 180 miles.

What is the standard form of the hyperbola that the receiver sits on if the transmitters behave as foci of the hyperbola?

2) 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?

Nov 21, 2019

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Here's the answer to the first question :

https://web2.0calc.com/questions/loran-question

Nov 21, 2019
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2) 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?

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

Nov 21, 2019