From the top of a tower 60m high, two ships are seen in a direction due south. The angle of depression of the ships from the top of the tower are 45degre and 30degree. Find the distance between the ships
Review your basic trig functions and you can easily see that the distance is
60cot30° - 60cot45°
To find the distance between the ships, we can use trigonometry and create a right triangle.
Let's consider the angle of depression of the ship closest to the tower, which is 45 degrees. Let A represent the top of the tower, B represent the ship closest to the tower, and C represent the ship farthest from the tower.
Now, we need to find the height of the tower, AB, and AC.
From the given information, we know that the height of the tower, AB, is 60m.
Next, we can use the tangent function to find the values of AB and AC.
In the triangle ABC, we have:
tan(45 degrees) = AB/AC
We know that tan(45 degrees) = 1, so we can rewrite the equation as:
1 = 60/AC
Simplifying, we find:
AC = 60m
Now, we can consider the angle of depression of the ship farthest from the tower, which is 30 degrees.
In the triangle ABC, we have:
tan(30 degrees) = AB/AC
We know that tan(30 degrees) = 1/sqrt(3), so we can rewrite the equation as:
1/sqrt(3) = 60/AC
Simplifying, we find:
AC = 60sqrt(3)/3 = 20sqrt(3)m
Therefore, the distance between the ships, BC, can be found by subtracting the values of AC:
BC = AC - AB = (20sqrt(3) - 60)m
Simplifying further, we get:
BC = 20sqrt(3) - 60 ≈ -9.95m
Since the distance between the ships cannot be negative, we can conclude that the distance between the ships is approximately 9.95m.
To find the distance between the two ships, we can use trigonometry and the concept of angles of depression.
First, let's draw a diagram to represent the situation:
30° 45°
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Tower (60m)
From the diagram, we can see that we have a right triangle formed between the top of the tower, one ship, and the base of the tower. Let's assume that the distance between the tower and the ship at a 30° angle of depression is x.
Now, we can use the trigonometric ratio tangent to find x:
tan(30°) = opposite/adjacent
tan(30°) = 60/x
Applying the same logic to the triangle formed by the tower, the other ship, and the base of the tower, we can also find the distance between the tower and the ship at a 45° angle of depression. Let's assume that the distance between the tower and this ship is y.
tan(45°) = opposite/adjacent
tan(45°) = 60/y
Now we have two equations with two unknowns:
1) tan(30°) = 60/x
2) tan(45°) = 60/y
We can solve this system of equations simultaneously to find the values of x and y. Dividing equation 1 by equation 2:
(tan(30°) / tan(45°)) = (60/x) / (60/y)
tan(30°) / tan(45°) = y / x
Next, substitute the values of the tangent of 30° and tangent of 45°:
(1/√3) / 1 = y / x
√3/3 = y / x
Cross-multiplying, we get:
√3 * x = 3y
Now substitute the value of y in equation 2 from equation 1:
tan(45°) = 60/((√3/3) * x)
1 = 60/((√3/3) * x)
(√3/3) * x = 60
Multiply both sides of the equation by 3/√3:
x = 60 * (3/√3)
Simplifying:
x = 60 * (√3/√3)
x = 60 * √3
Therefore, the distance between the ships is approximately 60 * √3 meters.