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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.
To find the equation of a hyperbola when the center is at the origin and the transverse axis is horizontal, we use the formula:
x^2 / a^2 - y^2 / b^2 = 1
where a = 55,000 km and c = 81,000 km. We can find b using the relationship b^2 = c^2 - a^2.
b^2 = 81000^2 - 55000^2
b^2 = 6561000000 - 3025000000
b^2 = 3536000000
b = √3536000000
b ≈ 59464.82 km
Now we can write the equation of the hyperbola:
x^2 / 55000^2 - y^2 / 59464.82^2 = 1
Simplifying this equation, we get:
x^2 / 3025000000 - y^2 / 3536000000 = 1
Therefore, the equation that models the path of the satellite in the form of a hyperbola is:
x^2 / 3025000000 - y^2 / 3536000000 = 1