The equation of a hyperbola with center at the origin, transverse axis along the x-axis, and distance from the center to a focus of c units is given by:
$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,$$
where b is the distance between the center and a vertex (related to a and c by the formula c^2 = a^2 + b^2).
For this particular hyperbola, with a = 55,000 km and c = 81,000 km, we can first solve for b using the formula c^2 = a^2 + b^2:
$$81,000^2 = 55,000^2 + b^2$$
$$6,561,000,000 = 3,025,000,000 + b^2$$
$$b^2 = 3,536,000,000$$
$$b = \sqrt{3,536,000,000}$$
$$b = 59,500 \text{ km}$$
Now we can plug in the values of a, b, and the center at the origin to get the equation of the hyperbola:
$$\frac{x^2}{55000^2} - \frac{y^2}{59500^2} = 1$$
$$\frac{x^2}{3,025,000,000} - \frac{y^2}{3,540,250,000} = 1$$
Thus, the equation that models the path of the satellite is:
$$\frac{x^2}{3025000000} - \frac{y^2}{3540250000} = 1$$
Find an equation that models the path of a satellite if its path is a hyperbola, a = 55,000 km, and c = 81,000 km. Assume that the center of the hyperbola is the origin and the transverse axis is horizontal...
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