Question
A weather balloon floats at a height of 15 kilometers above the surface of Earth. Earth's radius is approximately 6,371 kilometers. If the center of Earth is taken as the origin, which equation represents the weather balloon's position in a 2D plane?(1 point)
Responses
x2+y2=40,629,641
x squared plus y squared is equal to 40629641
x2+y2=40,780,996
x squared plus y squared is equal to 40780996
x2+y2=40,589,641
x squared plus y squared is equal to 40589641
x2+y2=40,398,736
Responses
x2+y2=40,629,641
x squared plus y squared is equal to 40629641
x2+y2=40,780,996
x squared plus y squared is equal to 40780996
x2+y2=40,589,641
x squared plus y squared is equal to 40589641
x2+y2=40,398,736
Answers
GPT-4o mini
To find the equation representing the weather balloon's position in a 2D plane, we first need to determine the distance from the center of the Earth to the weather balloon.
The radius of Earth is approximately 6,371 kilometers. The weather balloon is floating at a height of 15 kilometers above the surface. Therefore, the total distance from the center of the Earth to the weather balloon is:
\[
\text{Distance} = \text{Radius of Earth} + \text{Height of balloon} = 6,371 \text{ km} + 15 \text{ km} = 6,386 \text{ km}
\]
To convert this distance into meters (since the equations appear to use square kilometers):
\[
6,386 \text{ km} = 6,386,000 \text{ m}
\]
Now, the equation of a circle in the 2D plane centered at the origin is given by:
\[
x^2 + y^2 = r^2
\]
where \( r \) is the radius of the circle. We need to find \( r^2 \):
\[
r^2 = (6,386 \text{ km})^2 = 6,386^2 = 40,780,996 \text{ km}^2
\]
Therefore, the equation representing the weather balloon's position is:
\[
x^2 + y^2 = 40,780,996
\]
So the answer is:
**x² + y² = 40,780,996**
The radius of Earth is approximately 6,371 kilometers. The weather balloon is floating at a height of 15 kilometers above the surface. Therefore, the total distance from the center of the Earth to the weather balloon is:
\[
\text{Distance} = \text{Radius of Earth} + \text{Height of balloon} = 6,371 \text{ km} + 15 \text{ km} = 6,386 \text{ km}
\]
To convert this distance into meters (since the equations appear to use square kilometers):
\[
6,386 \text{ km} = 6,386,000 \text{ m}
\]
Now, the equation of a circle in the 2D plane centered at the origin is given by:
\[
x^2 + y^2 = r^2
\]
where \( r \) is the radius of the circle. We need to find \( r^2 \):
\[
r^2 = (6,386 \text{ km})^2 = 6,386^2 = 40,780,996 \text{ km}^2
\]
Therefore, the equation representing the weather balloon's position is:
\[
x^2 + y^2 = 40,780,996
\]
So the answer is:
**x² + y² = 40,780,996**