11-1 Lines that Intersect Circles
8. The International Space Station orbits Earth at an altitude of 240 miles. What is the distance from the space station to Earth's horizon to the nearest mile?
Sorry Steve, you made a major error with your units... the radius of the Earth is around 6378 km not 6378 miles.
You are trying to find the distance from the Space Station to the horizon, which would be the point of tangency. The tangent line from the Space Station will form a right angle with the center of the Earth. The radius of the Earth is 4000 miles, and the distance from the center of the earth to the Space Station is 4240 miles from the center of the Earth (radius + altitude).
Use Pythagorean Theorem:
radius of Earth squared + distance to horizon squared = distance from center of earth to Space Station squared.
4000^2 + x^2 = 4240^2
x^2 = 1977600
x = 1406.27
radius of earth: 6378
radius of orbit: 6618
draw a right triangle with legs 6378 and d, and hypotenuse 6618
6378^2 + d^2 = 6618^2
d^2 = 3119040
d = 1766 mi
Sorry Peter- you made a major error in your answer - but the question asks for miles, not kilometers.
Centimeters are cool too!
To calculate the distance from the International Space Station (ISS) to Earth's horizon, we need to consider the relationship between the Earth's radius and the altitude of the ISS.
Here's how you can calculate the distance:
1. Determine the radius of the Earth: The average radius of the Earth is approximately 3,959 miles.
2. Add the altitude of the ISS to the Earth's radius: 3,959 miles + 240 miles = 4,199 miles.
3. Calculate the distance from the ISS to the Earth's horizon: The distance can be found using the Pythagorean theorem, which states that for a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the Earth's radius and the distance from the ISS to the horizon form the two sides of the triangle.
Let's denote the distance from the ISS to the horizon as 'd'. The Earth's radius is 4,199 miles.
Using the Pythagorean theorem, we have the equation:
(4,199^2) = (d^2) + (3,959^2)
4. Simplify the equation and solve for 'd':
(4,199^2) - (3,959^2) = (d^2)
(17,632,801) - (15,672,081) = (d^2)
(1,960,720) = (d^2)
Now, calculate the square root of both sides to get 'd':
d = √(1,960,720)
d ≈ 1,399 miles
Therefore, the distance from the International Space Station (ISS) to Earth's horizon, to the nearest mile, is approximately 1,399 miles.