Question

The Kobe Port Tower is a hyperboloid structure in Kobe, Japan. This means that the shape is generated by rotating a hyperbola around its conjugate axis. Suppose the hyperbola used to generate the hyperboloid modeling the shape of the tower has an eccentricity of 19. If the tower is 8 meters wide at its narrowest point, determine an equation of the hyperbola used to generate the hyperboloid.

Answer 1

The equation of the hyperbola used to generate the hyperboloid is x^2/16 - y^2/56 = 1.

Step-by-step explanation:

To determine the equation of the hyperbola used to generate the hyperboloid modeling the shape of the tower, we need to use the information given. We know that the hyperbola has an eccentricity of 19 and the tower is 8 meters wide at its narrowest point. The width of the hyperbola is determined by the distance between the vertices, which is equal to 2a.

We can use the formula for the eccentricity of a hyperbola, which is e = c/a, where c is the distance from the center to each focus. Since the tower width is 8 meters, we can find a = 4.

Now we can calculate c using the equation e = c/a. Substituting the values, we get 19 = c/4. Multiplying both sides by 4 gives us c = 76.

Therefore, the equation of the hyperbola used to generate the hyperboloid is:

x^2/16 - y^2/56 = 1