The portion of the Earth that you can see from the hot air balloon is approximately 68.1 miles.
What is horizon ?
The horizon is the apparent boundary where the Earth's surface seems to meet the sky. It's the line that separates the visible portion of the Earth's surface from the portion that is not visible due to the curvature of the Earth. The horizon appears to be a straight line, but it is actually a curved line due to the spherical shape of the Earth. The distance to the horizon depends on the observer's height above the ground, as well as the radius of the Earth. The higher the observer, the farther away the horizon appears.
The distance from the balloon to the horizon is the same as the radius of the Earth plus the height of the balloon. So in this case, it's 1.5 + 4000 = 4001.5 miles.
Now we need to find the distance AC, which is the distance from the center of the Earth to point C. To do this, we can use the Pythagorean theorem:
AC² + BC² = (radius of Earth)²
AC² + (4001.5)² = (4000)²
AC² = (4000)² - (4001.5)²
AC ≈ 68.2 miles
Now we can find the distance BD using similar triangles. In the triangle ABD, we know that AB is the radius of the Earth, which is 4000 miles. We also know that AC is 68.2 miles, and we want to find BD. So we can set up the following proportion:
BD / AB = AC / AD
Solving for BD, we get:
BD = AB * (AC / AD)
To find AD, we can use the Pythagorean theorem again:
AD² = AB²+ BD²
AD² = (4000)²+ (BD)²
AD ≈ 4000.3 miles
Now we can plug in the values we have:
BD = AB * (AC / AD)
BD = 4000 * (68.2 / 4000.3)
BD ≈ 68.1 miles
So the portion of the Earth that you can see from the hot air balloon is approximately 68.1 miles. Rounded to the nearest tenth, the answer is 68.1 miles.
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