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Graph ( x + 2 ) 2 = −20 ( y 3 ) . Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.

Vertex: ( 2 , 3 ) ; Axis of symmetry: x = −2 ; Focus: ( 2 , 2 ) ; Directrix: y = 8 ; Endpoints of the latus rectum: ( 12 , 2 ) and ( 8 , 2 ) .

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Solving applied problems involving parabolas

As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry    are directed toward any surface of the mirror, the light is reflected directly to the focus. See [link] . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.

Reflecting property of parabolas

Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.

Solving applied problems involving parabolas

A cross-section of a design for a travel-sized solar fire starter is shown in [link] . The sun’s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds.

  1. Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane.
  2. Use the equation found in part (a) to find the depth of the fire starter.
Cross-section of a travel-sized solar fire starter
  1. The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form x 2 = 4 p y , where p > 0. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have p = 1.7.
    x 2 = 4 p y Standard form of upward-facing parabola with vertex (0,0) x 2 = 4 ( 1.7 ) y Substitute 1 .7 for  p . x 2 = 6.8 y Multiply .
  2. The dish extends 4.5 2 = 2.25 inches on either side of the origin. We can substitute 2.25 for x in the equation from part (a) to find the depth of the dish.
             x 2 = 6.8 y Equation found in part (a) . ( 2.25 ) 2 = 6.8 y Substitute 2 .25 for  x .            y 0.74   Solve for  y .

    The dish is about 0.74 inches deep.

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Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is placed 320 mm from the base.

  1. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the x -axis as its axis of symmetry).
  2. Use the equation found in part (a) to find the depth of the cooker.
  1. y 2 = 1280 x
  2. The depth of the cooker is 500 mm
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Access these online resources for additional instruction and practice with parabolas.

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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