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.
Since the transverse axis is horizontal, the equation of the hyperbola can be written as:
x^2 / a^2 - y^2 / b^2 = 1
where a = 55000 km and c = 81000 km. We can find b using the relationship between a, b, and c in a hyperbola:
c^2 = a^2 + b^2
b^2 = c^2 - a^2
b^2 = (81000)^2 - (55000)^2
b^2 = 6561000000 - 3025000000
b^2 = 3536000000
b = sqrt(3536000000)
b = 59429.82
Therefore, the equation of the hyperbola that models the path of the satellite is:
x^2 / (55000)^2 - y^2 / (59429.82)^2 = 1