EXAMPLE 4 Use a Hyperbola to Model a Real-World SituationSome telescopes use a hyperbolic mirror. Light is directed inward at thehyperbolic mirror. It is reflected through a hole in the parabolic mirroris shown on a coordinate plane. What equation can be used to represent thetoward a focus, where it can be captured by a detector. A hyperbolic mirrormirror? How far must the detector be placed from the mirror?Hyperbolic Secondary MirrorParabolic Primary MirrorFocalPointFmirror(6, 3)dfocus/(5,0) focusOne of the vertices is at (5, 0), so a = 5. A given point on the hyperbola is (6, 3).y²= 16²= 162 3252 6²36The foci are on the x-axis,so this is a horizontalhyperbola. Use the form935-2=1x²y²a²b²= 1.36b²-225=256²11b² = 225b² = 20.5The equation that models this hyperbolic mirror is - 20.5 = 1.The detector would have to be placed at the focus of the other branch of thehyperbola, at the point (-c, 0). Since c²= a² + b², the coordinates for thedetector are (-√45.5, 0) or about (-6.7, 0) which is about 11.7 units awayfrom the mirror.lightI 26. A telescope is made with a hyperbolic mirror,shown graphed on the coordinate plane.To assemble the telescope correctly, themanufacturer needs to know the distance tothe focus on the same side as the reflectiveside of the mirror. What is this distance?What equation represents the location of themirror? SEE EXAMPLE 48 16 24(12-50)(0-5)-16-24

Answered: Image /qna-images/answer/f2ff6c53-2887-490e-afac-ee4de641eb86.jpg

Answered: 26. A telescope is made with a…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Rational Functions And Conics

Section4.3: Conics

Problem 83E: When an airplane travels faster than the speed of sound, the sound waves form a cone behind the...

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Similar questions

Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road is 32 feet wide and 0.4 foot higher in the center than it is on the sides (see figure). (a) Write an equation of the parabola with its vertex at the origin that models the road surface. (b) How far from the center of the road is the road surface 0.1 foot lower than the center?
Sonic boom The position of a sonic boom caused by the faster-than-sound aircraft is one branch of the hyperbola y2x2=25 in the coordinate system shown. How wide is the hyperbola 5 miles from its vertex?
DISCUSS: Light from a Lamp The light from a lamp forms a lighted area on a wall, as shown in the figure. Why is the boundary of this lighted area a hyperbola? How can one hold a flashlight so that its beam forms a hyperbola on the ground?
When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. If the airplane is flying parallel to the ground, then the sound waves intersect the ground in a hyperbola with the airplane directly above its center, and a sonic boom is heard along the hyperbola. You hear a sonic boom that is audible along a hyperbola with the equation (x2/100)(y2/4)=1, where x and y are measured in miles. What is the shortesthorizontal distance you could be from the airplane?
The light bulb in a flashlight is at the focus of the parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation for a cross section of the flashlight’s reflector with its focus on the positive x-axis and its vertex at the origin.
The receiver of a parabolic satellite dish is at the focus of the parabola (see figure). Write an equation for a cross section of the satellite dish.
Beam Deflection A simply supported beam (see figure) is 64 feet long and has a load at the center. The deflection of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. (a) Write an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the center of the beam is the deflection 12 inch?
Think About It Write an equation whose graph is the bottom half of the hyperbola 9x254x4y2+8y+41=0.
Fluid Flow Water flowing from a horizontal pipe 48 feet above the ground has the shape of a parabola whose vertex 0,48 is at the end of the pipe (see figure). The water strikes the ocean at the point 103,0. (a) Find the standard form of the equation that represents the path of the water. (b) The pipe is moved 12 feet up. Find an equation that represents the path of the water. Find the coordinates of the point where the water strikes the ocean.

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