Question
A space station orbits Earth at a height of 250 kilometers. Earth's radius is 6,371 kilometers. If the center of Earth is considered the origin, what equation represents the station's orbit in a 2D plane?(1 point)
Responses
x2+y2=37,466,641
x squared plus y squared is equal to 37 comma 466 comma 641
x2+y2=43,837,641
x squared plus y squared is equal to 43 comma 837 comma 641
x2+y2=40,589,641
x squared plus y squared is equal to 40 comma 589 comma 641
x2+y2=40,652,141
Responses
x2+y2=37,466,641
x squared plus y squared is equal to 37 comma 466 comma 641
x2+y2=43,837,641
x squared plus y squared is equal to 43 comma 837 comma 641
x2+y2=40,589,641
x squared plus y squared is equal to 40 comma 589 comma 641
x2+y2=40,652,141
Answers
GPT-4o mini
To determine the equation representing the space station's orbit, we need to find the distance from the center of the Earth to the space station.
1. The radius of Earth is 6,371 kilometers.
2. The space station orbits at a height of 250 kilometers above the Earth's surface.
First, we calculate the total distance from the center of the Earth to the space station:
\[
\text{Total distance} = \text{Earth's radius} + \text{Height of the station} = 6,371 \, \text{km} + 250 \, \text{km} = 6,621 \, \text{km}
\]
Next, we need to express this in the form of an equation. The equation for a circle in a 2D plane, centered at the origin, is given by:
\[
x^2 + y^2 = r^2
\]
where \( r \) is the radius of the circle. Here, \( r = 6,621 \, \text{km} \).
Calculating \( r^2 \):
\[
r^2 = 6,621^2 = 43,839,961
\]
Now, we compare this value with the options given:
1. \( 37,466,641 \)
2. \( 43,837,641 \)
3. \( 40,589,641 \)
4. \( 40,652,141 \)
None of these match \( 43,839,961 \) exactly, but the closest option is:
**\( x^2 + y^2 = 43,837,641 \)**
This option is likely a rounding or simplification, making it the best choice based on the provided responses.
Thus, the answer is:
**\( x^2 + y^2 = 43,837,641 \)**
1. The radius of Earth is 6,371 kilometers.
2. The space station orbits at a height of 250 kilometers above the Earth's surface.
First, we calculate the total distance from the center of the Earth to the space station:
\[
\text{Total distance} = \text{Earth's radius} + \text{Height of the station} = 6,371 \, \text{km} + 250 \, \text{km} = 6,621 \, \text{km}
\]
Next, we need to express this in the form of an equation. The equation for a circle in a 2D plane, centered at the origin, is given by:
\[
x^2 + y^2 = r^2
\]
where \( r \) is the radius of the circle. Here, \( r = 6,621 \, \text{km} \).
Calculating \( r^2 \):
\[
r^2 = 6,621^2 = 43,839,961
\]
Now, we compare this value with the options given:
1. \( 37,466,641 \)
2. \( 43,837,641 \)
3. \( 40,589,641 \)
4. \( 40,652,141 \)
None of these match \( 43,839,961 \) exactly, but the closest option is:
**\( x^2 + y^2 = 43,837,641 \)**
This option is likely a rounding or simplification, making it the best choice based on the provided responses.
Thus, the answer is:
**\( x^2 + y^2 = 43,837,641 \)**