Hyperbola Equations in {x | -50 < x < 50}, {y | 0 < y < 20}

  • Thread starter yourmom98
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In summary: Ok. That is useful. Assume the curve touches the x-axis at x=-50 and +50 and reaches maximum y of 20 at x=0. You just have to determine the values for a and k (h=0 if it is to be centred around x=0). Let x=0 and y = 20 to work out value for k: (y-k)^2 = b^
  • #1
yourmom98
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Using a domain of {x | -50 < x < 50} and a range of {y | 0 < y < 20}, determine the following types of equations that you could use to model the curved arch.

The equation of a hyperbola in the form , where b = 10. The lower arm of the hyperbola would represent the arch.
((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=-1

i have 2 questions about this number one shouldn't the equations be

((x-h)^2)/(b^2)-((y-k)^2)/(a^2)=-1 because it should be a hyberbola that opens up and down?
and second how do i solve this?
 
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  • #2
yourmom98 said:
Using a domain of {x | -50 < x < 50} and a range of {y | 0 < y < 20}, determine the following types of equations that you could use to model the curved arch.

The equation of a hyperbola in the form , where b = 10. The lower arm of the hyperbola would represent the arch.
((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=-1

i have 2 questions about this number one shouldn't the equations be

((x-h)^2)/(b^2)-((y-k)^2)/(a^2)=-1 because it should be a hyberbola that opens up and down?
and second how do i solve this?
I think you are missing part of the problem. Does the question give you a value for C, the distance from the origin to the foci?

AM
 
  • #3
nope it does say that the graph is supposed to be a curved arch that will have horizontal span of 100m and a maximum height of 20m.
 
  • #4
yourmom98 said:
nope it does say that the graph is supposed to be a curved arch that will have horizontal span of 100m and a maximum height of 20m.
Ok. That is useful. Assume the curve touches the x-axis at x=-50 and +50 and reaches maximum y of 20 at x=0. You just have to determine the values for a and k (h=0 if it is to be centred around x=0).

Let x=0 and y = 20 to work out value for k: [itex](y-k)^2 = b^2[/itex]
Then let x = 50 and y=0 to get the value for a: [itex]b^2x^2 - a^2(y-k)^2 = -a^2b^2[/itex]

AM
 

What is a hyperbola?

A hyperbola is a type of conic section, created by intersecting a cone with a plane that is parallel to one of the cone's sides. It is a symmetrical curve that consists of two branches that are mirror images of each other.

What are the key components of a hyperbola equation?

The key components of a hyperbola equation are the center, the vertices, the foci, and the asymptotes. The center is the point around which the hyperbola is symmetrical. The vertices are the points where the hyperbola intersects with its transverse axis. The foci are the two fixed points that determine the shape and size of the hyperbola. The asymptotes are the lines that the hyperbola approaches but never crosses.

How do you graph a hyperbola?

To graph a hyperbola in the given domain and range, you can use the key components of the equation. Plot the center, vertices, and foci on the coordinate plane. Then, draw the asymptotes passing through the vertices and center. Finally, sketch the two branches of the hyperbola, using the asymptotes as a guide.

What is the difference between a horizontal and a vertical hyperbola?

A horizontal hyperbola has its transverse axis along the x-axis, while a vertical hyperbola has its transverse axis along the y-axis. This means that the equations for these hyperbolas will have different forms, but their key components will still be the same.

How are hyperbolas used in real life?

Hyperbolas have many real-life applications, including in satellite communication, optics, and the study of planetary orbits. They are also used in economics and finance to model supply and demand curves and in engineering to design rocket and missile trajectories.

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