On the bridge of a ship at sea, the captain asked the new, young officer standing next to him to determine the distance to the horizon. The officer took pencil and paper, and in a few moments came up with an answer. On the paper he had written the formula d = 3.6 square root h. Show that this formula is a good approximation of the distance, in kilometers, to the horizon, if h is the height, in meters, of the observer above the water. (Assume the radius of the earth to be 6500 km.) If the bridge was 27m. above the water, what was the distance to the horizon?

Question

On the bridge of a ship at sea, the captain asked the new, young officer standing next to him to determine the distance to the horizon. The officer took pencil and paper, and in a few moments came up with an answer. On the paper he had written the formula d = 3.6 square root h. Show that this formula is a good approximation of the distance, in kilometers, to the horizon, if h is the height, in meters, of the observer above the water. (Assume the radius of the earth to be 6500 km.) If the bridge was 27m. above the water, what was the distance to the horizon?

I understand how to find the distance by just plugging it into the formula, but I would like to know the deviation of the formula. Here is a picture of the problem: gyazo(DOT)com/2acde0ce3958354b207df75329356b8a

Answer 1

no deviation to that formula. Maybe you'd like the derivation...

Draw a radius r, extended by h.
Draw a second radius r.
Connect the two endpoints, and that is the distance to the horizon.

d = √((r+h)^2-r^2) = √(2rh+h^2)

That's the distance in km. Usually h, in meters is very much smaller than r, so

d ≈ √(2rh)
Now, using km for everything,
d ≈ √(2*6500*h/1000) = √(13h) = 3.6√h

Answer 2

To determine the deviation of the formula d = 3.6 √h for calculating the distance to the horizon, we can compare it with the more precise formula. The more precise formula for calculating the distance to the horizon, based on the curvature of the Earth, is given by d = √(2Rh+h^2), where R is the radius of the Earth.

In this scenario, the radius of the Earth is assumed to be 6500 km.

Now, let's substitute the given height of the observer above the water (h = 27 m) into both formulas and calculate the distances.

1. Using the simplified formula d = 3.6 √h:
d = 3.6 √27
d ≈ 3.6 × 5.196
d ≈ 18.7056 km

2. Using the precise formula d = √(2Rh+h^2):
R = 6500 km
d = √(2 × 6500 × 27 + 27^2)
d = √(351000 + 729)
d = √351729
d ≈ 592.82 km

To find the deviation between the two formulas, we can subtract the result obtained from the simplified formula from the result obtained from the precise formula:

Deviation = Precise formula result - Simplified formula result
Deviation = 592.82 km - 18.7056 km
Deviation ≈ 574.1144 km

Therefore, the deviation of the simplified formula for the distance to the horizon is approximately 574.1144 km.