CP PC Conic Applications

CP PC Conic Applications

Name: ______3-10-14

Example #1:

The semielliptical arch in a concrete bridge must have a clearance of 12 feet above the water and span a distance of 40 feet. Find the equation of the ellipse after inserting a coordinate plane with the center of the ellipse at the origin and the major axis on the x-axis. How much clearance above the water is there 5 feet from the bank?

Example #2:

A ship is traveling on a course parallel and 60 miles form a straight shoreline. Two transmitting stations, S1 and S2, are located 200 miles apart on the shoreline. By timing radio signals from the stations, the ship's navigator determines that the ship is between the two stations and 50 miles closer to S2 than to S1. Find the distance from the ship to each station.

Example #3:

A designer of a 200-foot diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 100 feet above the vertex.

a.  Find the equation of the parabola using the axis of the parabola as the y-axis and the vertex at the origin.

b.  Determine the depth of the parabolic reflector.

Assignment:

1)  Suppose an earthquake can be felt up to 80 miles from its epicenter. You are located at a point 60 miles west and 45 miles south of the epicenter. Do you feel the earthquake? If so, how many miles due south would you have to travel to be out of range of the earthquake?

2)  The filament (thin wire that glows when electricity passes through it) of a car headlight is at the focus of a parabolic reflector, which sends light out in a straight beam. Given that the filament is 1.5 inches from the vertex,

a)  find the equation for the cross section of the reflector.

b)  if the reflector is 7 inches wide, how deep is it?

3)  On the sidelines of each of its televised football games, the FBTV network uses a parabolic reflector with a microphone at the reflector’s focus to capture the conversations among players on the field. If the parabolic reflector is 3 feet across and 1 foot deep, where should the microphone be placed?

4)  Andrew and David live exactly 4 miles apart. While on the phone at their homes, David hears thunder from a lightning bole outside and 10 seconds later at Karl’s house through the phone. Sounds travels about 1 mile every 5 seconds.

a)  On a coordinate plane, assume that Andrew is located at (-2, 0) and David is located at (2, 0). Determine the equation for the horizontal hyperbola on which the lightning is located.

b)  Which branch of the hyperbola corresponds to the places where the lightning bolt may have struck?

5) Statuary Hall is an elliptical room in the U.S. Capitol in Washington, D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear a whisper spoke by a person standing at the other focus. Statuary Hall is 46 feet wide and 97 feet long.

a) Find an equation that models the shape of the room.

b) How far apart are the two foci?

c) What is the area of the floor of the room?

( where and are the lengths of the semi-major and semi-minor axes)

6)  A lithotripter is formed by rotation the portion of an ellipse below the minor axis around the major axis. The lithotripter is 20 centimeters wide and 16 centimeters deep. If the ultrasound source is positioned at one focus of the ellipse and the kidney stone at the other, then all the sound waves will pass through the kidney stone. How far from the kidney stone should point V on the base of the lithotripter be positioned to focus the sound waves on the kidney stone?

7)  A hyperbolic mirror can be used to take panoramic photographs. A camera is pointed toward the vertex of the mirror and is positioned so that the lens is at one focus of the mirror. An equation for the cross section of the mirror is where and are measured in inches. If you place a camera with its lens at the focus, how far is the lens from the vertex of the mirror?