High School

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### Task

1. A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 2 inches from the vertex. If the depth is to be 3 inches, what is the diameter of the headlight at its opening? Give an exact answer, not a decimal. Make sure to simplify. (12 pts)

2. The mirror of a flashlight is a paraboloid of revolution. Its diameter is 6 cm, and its depth is 2 cm. How far from the vertex should the filament of the light bulb be placed for the flashlight to have its beam run parallel to the axis of symmetry? (Note: picture not to scale. Give exact answer, not a decimal.) (12 pts)

3. Ellipses have interesting reflection properties. If a source of light or sound is placed at one focus, the waves reflect off the ellipse and concentrate on the other focus. This is the principle behind whispering galleries, where a person standing at one focus can whisper and be heard by a person standing a distance away at the other focus.

- Jim, standing at one focus of a whispering gallery, is 1 foot from the nearest wall. His friend, Leonard, is standing at the other focus 120 feet away. Round answers to the nearest tenth place.

11. What is the length of this whispering gallery? (6 pts)
12. How high is its elliptical ceiling at the center? (12 pts)

4. Hyperbolas have reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would theoretically pass through its rear focus, it is reflected through the front focus. This property, and that of the parabola, were used to develop the Cassegrain telescope in 1672. The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, reflected toward the common focus, and then reflected by the hyperbolic mirror through the opening to its front.

13. If the equation of the hyperbolic mirror in a Cassegrain telescope is \(\frac{x^2}{25} - \frac{y^2}{144} = 1\) and the focal length of the parabola is 15, find the equation of the parabola. (Hint: the equation of a parabola is \((x-h)^2 = 4p(y-k)\) for a parabola opening up/down or \((y-k)^2 = 4p(x-h)\) for a parabola opening left/right, where \(p\) is the focal length and \((h,k)\) are the coordinates of the vertex.)

Answer :

  1. For the sealed-beam headlight, using the paraboloid equation y = x²/4f with f=2 inches, the diameter at the opening is 4√6 inches.
  2. The flashlight's filament should be placed at a distance of 1.125 cm from the vertex for the beam to be parallel.
  3. The whisper gallery's length is 121 feet, and the equation of the parabolic mirror in the Cassegrain telescope is x² =60y.

1) To find the diameter of the headlight at its opening, we will use the formula of the paraboloid of revolution which is given by y = x²/(4f), where f is the distance between the vertex and the focus, in this case, f=2 inches. In order to find the radius, we look at the value of x when the paraboloid reaches its depth (in this case, y=3 inches). Solving this equation for x when y=3 and f=2, we get x= √(4 * 2 * 3)= 4√(6) as the radius. The diameter is twice the radius, so it will be 4√(6) inches.

2) The focus of the mirror of the flashlight, where the filament of the light bulb should be placed, can be found using the paraboloid equation x^2 = 4fy. Given that the radius r is the diameter divided by 2 we get r=3 cm, and the depth is y=2 cm. Substituting these values into the paraboloid equation we find that the focus f will be (r²) / (4y) = (3²) / (4*2) = 9/8= 1.125 cm.

3) The whisper gallery forms an elliptical shape where Jim to the wall to Leonard forms the major axis (2a) of the ellipse. The distance from Jim to the wall is 1 foot and from Jim to Leonard is 120 feet, therefore, the length of the gallery, which is the same as the major axis, is 1+120= 121 feet.

Concerning the Cassegrain telescope and given the equation of the hyperbolic mirror 1/(144*x²) - 1/(25*y²) = 1, along with the focal length (p) of the parabolic mirror of 15, we can use the standard form of the parabolic mirror equation, (x-h)²=4p(y-k), to find its equation. A parabola that opens upwards has an equation of the form x² = 4p*y, and since the parabolic and hyperbolic mirror share the same focus, h = k. Given that the focus p = 15, the equation of the parabolic mirror will be x² = 60y.

To learn more about paraboloid visit :

https://brainly.com/question/29635857

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