on a lake without any waves, how far from the shore is a sailboat when the top of 10 m mast disappears from view of a person whose eye height is one m above the water's edge
To determine the distance from the shore to the sailboat when the top of the 10 m mast disappears from view, we can use basic trigonometry.
Let's denote the distance from the shore to the sailboat as "x". We also know that the height of the observer's eye is 1 m, and the height of the mast is 10 m.
In this case, we have a right triangle formed by the observer, the sailboat, and the top of the mast. The observer's eye is at the bottom vertex, and the top of the mast is at the top vertex of the triangle.
From the triangle, we can setup the following equation:
tan(θ) = opposite/adjacent
In this case, the opposite side is the height of the mast (10 m), and the adjacent side is the distance from the shore to the sailboat (x).
We can substitute the values into the equation:
tan(θ) = 10/x
To find the value of the angle θ, we can use the inverse tangent function to get:
θ = arctan(10/x)
Since we know that the observer's eye is 1 m above the water's edge, the total height of the observer is 1 m + 10 m (mast height) = 11 m.
When the observer's line of sight is parallel to the water's surface, the observer will be looking horizontally. In this case, the triangle formed is a right triangle, with the angle θ being 90 degrees.
We can set the two equations equal to each other:
θ = arctan(10/x) = 90 degrees
Since the tangent of 90 degrees is undefined, we can solve the equation by finding the limit as x approaches infinity.
lim x->infinity (arctan(10/x)) = 90 degrees
As x approaches infinity, the angle approaches 90 degrees. Therefore, the sailboat is infinitely far from the shore when the top of the mast disappears from the observer's view.
In other words, the sailboat will never disappear from view completely as long as the observer is standing 1 m above the water's edge.
To find out how far the sailboat is from the shore, we can use a concept called the line of sight. The line of sight is an imaginary straight line that connects the person's eye with the top of the mast on the sailboat. Since the person's eye height is one meter above the water's edge, the height of their line of sight will be the sum of their eye height and the height of the mast.
In this case, the height of the line of sight is 10 meters (mast height) + 1 meter (eye height) = 11 meters.
Now, we can imagine a right-angled triangle with the line of sight as one of the sides. The other side of the triangle will represent the distance from the person to the sailboat, and the hypotenuse will represent the distance from the person to the sailboat plus the distance from the sailboat to the shore.
Since the sailboat is on a lake without any waves, we can assume that the water's surface is flat. Therefore, we can use the Pythagorean theorem to find the distance from the person to the sailboat.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we have:
Distance^2 + 11^2 = Distance^2
Simplifying the equation:
Distance^2 + 121 = Distance^2
Subtracting Distance^2 from both sides:
121 = 0
This equation is not solvable, which means that the sailboat cannot disappear from view in this scenario. However, there could be an error or missing information, so please double-check the given values.
The distance D to the horizon in km by an observer at H m from the ground is given approximately by
D=3.56√H
The visible distance between two objects at H1 and H2 away from (the same) horizon is thus given by the sum of the distances, namely:
D=3.56(√H1+√H2)
=14.83 km.
See
http://en.wikipedia.org/wiki/Horizon
Follow the derivation of the formula to understand the geometry behind it.