The radius r of the Earth is 6400 km. Find the distance d to the horizon that a pilot can see on a clear day when flying at a height h of 5 km. Round your answer to the nearest hundredth km
The Correct Answer and Explanation is:
To solve this problem, the Pythagorean Theorem is applied to the right triangle formed by the Earth’s radius rrr, the pilot’s height above the Earth hhh, and the distance ddd to the horizon. The relevant triangle has sides:
- One leg: rrr (Earth’s radius)
- Hypotenuse: r+hr + hr+h (distance from Earth’s center to the pilot)
- Other leg: ddd (distance to horizon)
Using the Pythagorean Theorem:(r+h)2=r2+d2(r + h)^2 = r^2 + d^2(r+h)2=r2+d2
Solve for ddd:d2=(r+h)2−r2d^2 = (r + h)^2 – r^2d2=(r+h)2−r2
Substitute the values:
- r=6400r = 6400r=6400 km
- h=5h = 5h=5 km
d2=(6400+5)2−(6400)2d^2 = (6400 + 5)^2 – (6400)^2d2=(6400+5)2−(6400)2d2=(6405)2−(6400)2d^2 = (6405)^2 – (6400)^2d2=(6405)2−(6400)2
Now compute:(6405)2=41024025(6405)^2 = 41024025(6405)2=41024025(6400)2=40960000(6400)^2 = 40960000(6400)2=40960000d2=41024025−40960000=64025d^2 = 41024025 – 40960000 = 64025d2=41024025−40960000=64025
Then take the square root:d=64025≈252.99d = \sqrt{64025} \approx 252.99d=64025≈252.99
Final Answer: 252.99\boxed{252.99}252.99 km
Explanation (300 words):
In this scenario, the goal is to determine the distance to the horizon from a given height above the Earth’s surface. This type of geometric problem involves visualizing the Earth as a circle and applying the Pythagorean Theorem to a triangle formed by the Earth’s radius, the observer’s height, and the line of sight to the horizon.
The Earth’s radius acts as one leg of a right triangle, and the observer’s distance from the Earth’s center (radius plus height) serves as the hypotenuse. The distance to the horizon represents the other leg. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Using this principle, the equation is set up as (r+h)2=r2+d2(r + h)^2 = r^2 + d^2(r+h)2=r2+d2, where ddd is the desired distance.
Substituting the values for r=6400r = 6400r=6400 km and h=5h = 5h=5 km, the difference of squares yields d2=(6405)2−(6400)2d^2 = (6405)^2 – (6400)^2d2=(6405)2−(6400)2. Calculating both squares and subtracting them results in a value for d2d^2d2, which is then square-rooted to find the linear distance.
This type of calculation is common in aviation and geodesy, providing crucial visibility range estimates. The assumption in this problem is that the Earth is perfectly spherical and atmospheric refraction is negligible, which simplifies real-world conditions for clarity and learning purposes.
