Find an equation that models the paths of a satellite if its path is a hyperbola, a=45,000 km, and c=71,000 km. Assume that the center of the hyperbola is the origin and the transverse axis is horizontal.
Since the hyperbola is centered at the origin and the transverse axis is horizontal, the equation has the form:
x^2/a^2 – y^2/b^2 = 1
where a is the distance from the origin to either vertex, and b is the distance from the origin to either foci. We are given that a = 45,000 km, so:
x^2/45,000^2 – y^2/b^2 = 1
To find b, we use the relationship c^2 = a^2 + b^2, where c is the distance from the origin to either focus. We are given that c = 71,000 km, so:
71,000^2 = 45,000^2 + b^2
b^2 = 71,000^2 – 45,000^2
b^2 = 2,696,000,000 km^2
Now we can substitute this value of b^2 into the equation:
x^2/45,000^2 – y^2/2,696,000,000 = 1
This equation models the paths of the satellite if its path is a hyperbola, with a distance of 45,000 km from the origin to either vertex and 71,000 km from the origin to either focus.
To find an equation that models the path of the satellite, we can use the equation of a hyperbola centered at the origin with a horizontal transverse axis:
𝑥^2 / 𝑎^2 - 𝑦^2 / 𝑏^2 = 1
where:
- 𝑎 is the distance from the center to each vertex
- 𝑏 is the distance from the center to each co-vertex
In this case, 𝑎 is given as 45,000 km, and 𝑐 is given as 71,000 km. Since the hyperbola is centered at the origin, the distances from the center to each focus is also 𝑐.
To find 𝑏, we can use the relationship between 𝑎, 𝑏, and 𝑐 in a hyperbola:
𝑐^2 = 𝑎^2 + 𝑏^2
Substituting the given values:
(71,000 km)^2 = (45,000 km)^2 + 𝑏^2
Simplifying:
5,041,000,000 km^2 = 2,025,000,000 km^2 + 𝑏^2
Subtracting 2,025,000,000 km^2 from both sides:
3,016,000,000 km^2 = 𝑏^2
Taking the square root of both sides:
√(3,016,000,000 km^2) = √(𝑏^2)
√(3,016,000,000 km^2) = 𝑏
√(3016) km = 𝑏
So, 𝑏 is approximately 54.915 km.
Now we can substitute the values of 𝑎 and 𝑏 into the equation of the hyperbola:
𝑥^2 / (45,000 km)^2 - 𝑦^2 / (54.915 km)^2 = 1
Simplifying:
𝑥^2 / 2,025,000,000 km^2 - 𝑦^2 / 3,016 km^2 ≈ 1
Therefore, the equation that models the path of the satellite is:
𝑥^2 / 2,025,000,000 km^2 - 𝑦^2 / 3,016 km^2 ≈ 1