Question

An engineer is designing a bearing with a groove having edges in the shape of a hyperbola. A coordinate system has been set up with each unit representing one millimeter. The closest the bottom of the groove comes to the center of the bearing on the coordinate system is 16 millimeters. If the groove has edges that follow the asymptotes y=3.5x and y=−3.5x, find an equation for the hyperbola that can be used to model the edges of the groove.

Assume the hyperbola is vertical, and round your a and b values to the nearest hundredth place if necessary.

Answer 1

The equation for the hyperbola with given asymptotes y=3.5x and y=-3.5x is y=-x-16
`x^2`

Step-by-step explanation:

The equation for a hyperbola with vertical asymptotes y = 3.5x and y = -3.5x can be written in the form
`y = ax + bx^2`g this to the equation of a hyperbola, we can determine the values of a and b. The slope of the graph, which is equal to -a/b, is given by the ratio of the coefficients of x in the asymptotes equations. In this case, it is -3.5/3.5 = -1. Therefore, a = -1. The value of b can be determined by substituting the coordinates of a point on the hyperbola into the equation and solving for b. Using the given point (0, 16), we get
`16 = -b(0) + (0)^2` equation gives us b = -16. Therefore, the equation for the hyperbola is y = -x - 16
`x^2`

Answer 2

The equation for the hyperbola that models the edges of the groove is (y - 0)² / (16)² - (x - 0)² / (3.5*16)² = 1.

Step-by-step explanation:

To find the equation of the hyperbola that models the edges of the groove, we can use the standard form of the hyperbola equation, which is (y - k)² / a² - (x - h)² / b² = 1 for a vertical hyperbola. By analyzing the given information, we can determine the values of k, h, a, and b. The given asymptotes, y = 3.5x and y = -3.5x, intersect at the origin (h, k) = (0, 0). The distance between the bottom of the groove and the center of the bearing, which is the value of a, is 16 millimeters. As for the value of b, it can be determined by considering the distance between the edges of the groove and the center of the bearing, which is 3.5 times the corresponding x-coordinate. Therefore, b = 3.5 * a. Substituting these values into the standard form of the hyperbola equation, we get (y - 0)² / (16)² - (x - 0)² / (3.5*16)² = 1.