If you are in a hot-air balloon at a height of 100 meters, you can expect to see ____ km. Round your answer to the nearest kilometer and enter only the number.
if the earth's radius is r, then at height h, the distance d you can see is
r^2 + d^2 = (r+h)^2
However, that is just the distance from the balloon to the horizon. The actual distance seen along the earth is the part of the circumference subtended by the angle t, where
cos(t) = r/(r+h)
the distance measured along the surface of the earth is then
d = r*arccos(r/(r+h))
don't forget to express r and h in the same units, and use many significant digits, as h is very small compared to r.
To determine how far you can expect to see from a hot-air balloon at a height of 100 meters, we need to make some assumptions and calculations.
The distance you can see from a specific height depends on various factors such as atmospheric conditions, terrain, and the height of any obstacles between you and the horizon. However, as a rough estimate, we can use the formula for the distance to the horizon, which assumes a perfect spherical Earth with no obstacles:
Distance to the horizon (in kilometers) = √(2 * height (in meters) * radius of the Earth (in kilometers))
The radius of the Earth is approximately 6,371 kilometers.
Let's calculate the distance to the horizon at a height of 100 meters using this formula:
Distance to the horizon = √(2 * 100 * 6,371)
Distance to the horizon ≈ √(1,274,200)
Distance to the horizon ≈ 1,128 kilometers
Rounding this answer to the nearest kilometer, we can expect to see approximately 1,128 kilometers from a hot-air balloon at a height of 100 meters.