from the top of a light house the angle of depression of a ship sailing towards it was found to be 30 degrees . after 10 seconds the angle of depresion changes to 60 degrees. assuming that the ship is sailing at uniform speed. find how much time it will take to reach the light house?
Locate points on the ocean:
Let the point at 30 degrees be A
Let the point at 60 degrees be B
Let the base of the lighthouse be point C
Let the height of the lighthouse be h
h/BC = tan 60 = √3
h/AC = tan 30 = 1/√3
√3 BC = 1/√3 AC
3BC = AC
That means AC = AB + BC
SO, 3BC = AB + BC
So, AB = 2BC
AB was traveled in 10 seconds.
So, BC will take only 5 seconds to cover.
Watch out for those rocks!
To solve this problem, we can use trigonometric principles. Let's assume the height of the lighthouse is "h" and the distance between the ship and the base of the lighthouse is "d."
From the top of the lighthouse, the angle of depression to the ship is 30 degrees. This means that the tangent of the angle, tan(30 degrees), is equal to the height of the lighthouse divided by the distance to the ship:
tan(30 degrees) = h / d
After 10 seconds, the angle of depression changes to 60 degrees. Using the same logic as before, we can say that:
tan(60 degrees) = h / (d + 10s)
Since the ship is sailing at a uniform speed, we can assume that the distance traveled by the ship in 10 seconds is the same as the distance from the base of the lighthouse to the ship at the start. Therefore, we can rewrite the second equation as:
tan(60 degrees) = h / d
Now, we have two equations:
1. tan(30 degrees) = h / d
2. tan(60 degrees) = h / d
We can solve these equations simultaneously to find the values of "h" and "d."
tan(30 degrees) = h / d
1/sqrt(3) = h / d
(d * 1/sqrt(3)) = h
tan(60 degrees) = h / d
sqrt(3) = h / d
(d * sqrt(3)) = h
From the above two equations:
d * 1/sqrt(3) = d * sqrt(3)
1/sqrt(3) = sqrt(3)
This equation is not possible. Hence, we cannot find the values of "h" and "d" simultaneously.
Given that the distance traveled by the ship is constant, we can calculate the time it takes for the ship to reach the lighthouse using the equation:
time = distance / speed
Since we don't have the speed of the ship, we cannot determine the time it will take for the ship to reach the lighthouse with the given information.
To solve this problem, we can use trigonometry and a bit of algebra.
Let's denote the distance between the ship and the lighthouse by d (in units such as meters) and the speed of the ship by s (in units such as meters per second). We want to find the time it will take for the ship to reach the lighthouse, which we'll call t.
From the top of the lighthouse, the angle of depression to the ship changes from 30 degrees to 60 degrees after 10 seconds. This means that during these 10 seconds, the ship has moved a certain distance closer to the lighthouse.
First, let's consider the situation when the angle of depression is 30 degrees. We can use the tangent function to relate the angle of depression to the distance between the ship and the lighthouse:
tan(30 degrees) = d / x, where x is the distance between the lighthouse and the ship at the time when the angle of depression is 30 degrees.
Since we know that the tangent of 30 degrees is equal to 1/sqrt(3), we can rewrite the equation as:
1/sqrt(3) = d / x
Next, let's consider the situation when the angle of depression is 60 degrees, after 10 seconds. At this point, the ship has moved closer to the lighthouse, so the distance between them is now x - s * 10 (since the ship is moving at a speed of s meters per second).
Using the same logic as before, but now using the tangent of 60 degrees (which is sqrt(3)), we can set up another equation:
sqrt(3) = d / (x - s * 10)
Now, we have a system of two equations with two unknowns (d and x). We can solve this system of equations to find the values of d and x.
Let's first solve the first equation for x:
x = d * sqrt(3)
Now substitute this value of x into the second equation:
sqrt(3) = d / (d * sqrt(3) - s * 10)
Multiply both sides by d * sqrt(3) - s * 10 to eliminate the denominator:
sqrt(3) * (d * sqrt(3) - s * 10) = d
Simplifying, we get:
3d - 10 * sqrt(3) = d
2d = 10 * sqrt(3)
d = 5 * sqrt(3)
From here, we can substitute the value of d back into the equation x = d * sqrt(3) to find x:
x = 5 * sqrt(3) * sqrt(3)
x = 5 * 3
x = 15
Now that we know x, we can calculate t by dividing the distance x by the speed of the ship:
t = x / s
t = 15 / s
Therefore, the time it will take for the ship to reach the lighthouse is 15/s seconds.