A lifeguard was staring at a ship in the distance, and watched the smokestack as the ship went out to sea. The lifeguard's eye level is 4 meters above sea level. If the height of the smoke stack above water is 30 meters and the radius of earth is about 6400km, how far off from the shore would the ship be when the lifeguard who loose sight of it.
how should I start this?
I visualize a circle with radius 6400 and the ship at a point P.
let the eyes of the lifeguard be Q.
then Q is 6400.004 km from the centre of our earth circle.
let PQ = x
x^2 + 6400^2 = 6400.004^2
I got x = appr. 7.155 km
the answer is suppose to be 2675.3
..
To solve this problem, we will use the concept of curvature of the Earth and the line of sight. The key idea is that as the ship moves away from the shore, it will eventually disappear from the line of sight of the lifeguard due to the Earth's curvature.
To start solving the problem, we need to calculate the distance at which the ship would reach the horizon and become invisible to the lifeguard's line of sight. Here's how you can approach it:
1. Identify the known values:
- The height of the lifeguard above sea level = 4 meters
- The height of the smoke stack above water = 30 meters
- The radius of the Earth = 6400 kilometers (or 6,400,000 meters)
2. Use the Pythagorean theorem to find the distance from the lifeguard's eye level to the top of the smokestack:
- Distance = √(height of lifeguard)^2 + (height of smoke stack)^2
- Distance = √(4^2 + 30^2)
- Distance = √(16 + 900)
- Distance = √916
- Distance ≈ 30.266 meters
3. Now, we need to determine the distance at which the ship would disappear beyond the horizon. We can assume the ship is a speck, and only the top of the smokestack is visible. The distance to the horizon can be calculated using the formula for the horizon distance:
- Horizon distance = √(2 * radius of Earth * height of lifeguard)
- Horizon distance = √(2 * 6,400,000 * 4)
- Horizon distance = √(51,200,000)
- Horizon distance ≈ 7,159 meters
4. Finally, subtract the distance from the lifeguard's eye level to the top of the smokestack from the horizon distance to find out how far off from the shore the ship would be when the lifeguard loses sight of it:
- Final distance = Horizon distance - Distance
- Final distance ≈ 7,159 - 30.266
- Final distance ≈ 7,128.734 meters
Therefore, the ship would be approximately 7,128.734 meters (or about 7.13 kilometers) off from the shore when the lifeguard loses sight of it.