Math

posted by .

Cooling towers for nuclear reactors are often constructed as hyperboloids of one sheet because of the structural stability of that surface. Suppose all horizontal cross sections are circular, with a minimum radius of 200 feet occurring at a height of 600 feet. The tower is to be 800 feet tall with a maximum cross-sectional radius of 300 feet. Find the equation of the surface.

  • Math -

    Make the axis of symmetry the y axis and the minimum cross section be the y = 0 plane. The general equation for the hyperbola that generates the surface is

    x^2/a^2 - y^2/b^2 = 1

    where a is the minimum radial distance from the y axis, which in this case is 200 feet. To get b, require that x = 300 when y = 200 feet. (That is, y = 200 feet above the "throat" where x = 200.
    (300/200)^2 - (200/b)^2 = 1
    (200/b)^2 = 1.25
    b/200 = sqrt(4/5)
    b = 178.9 ft

    (x/200)^2 - (y/178.9)^2 = 1
    The domain of y is -600 to +200

  • Math -

    However, the equation is for a hyperboloid of one sheet, not a hyperbola. I need to find a equation that follows the general equation of a hyperboloid of one sheet: ax^2 + by^2 - cz^2 where a,b,c are > 0.

  • Math -

    The hyperbola that I described, when rotated about the y axis, in an x,y,z coordinate system. becomes the hyperboloid of one sheet. I leave the rest up to you

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math

    Consider the right circular cone shown. If the radius of the circular base remains the same and the height varies, what are the minimum surface area and minimum volume the cone can have?
  2. math

    Consider the right circular cone shown. If the radius of the circular base remains the same and the height varies, what are the minimum surface area and minimum volume the cone can have?
  3. math

    Consider the right circular cone shown. if the radius of the circular base remains the same and the height varies, what is the minimum surface area and minimum volume the cone can have?
  4. Calc 3

    A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 300 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation describing the shape …
  5. Math

    A cone is constructed by cutting a sector from a circular sheet of metal with radius 20cm. The cut sheet is then folded up and welded. Find the radius and height of the cone with the maximum volume that can be formed in this way.
  6. calculus

    A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 260 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation for the tower. (Assume …
  7. calculus

    A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 260 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation for the tower. (Assume …
  8. Math 117b

    describes h(x), the height of a part of a rollercoster track,where x is the horizontal distance in feet from the center of this section of the track. The towers that support this part of the track are the same height and are 150 feet …
  9. math

    the function h(x)=1/98x^2 describes h(x), the height of part of a roller coaster track, where x is the horizontal distance in feet from the center of this section of the track. The towers that support this part of the track are the …
  10. Calculus III

    Topic: Quadric surfaces. I have two homework problems, and they are quite a doozy for me. So the point is, I am given details of a quadric surface on the Cartesian coordinate system, and thus have to use the data to construct an equation …

More Similar Questions