To solve this problem, we can use some basic trigonometry. The situation involves a ship at a height of 10 meters above the water, with angles of elevation and depression to a hill. Let's break this down step by step to find both the distance to the hill and its height.
Understanding the Angles
We have two angles to consider:
- The angle of elevation to the top of the hill is 60 degrees.
- The angle of depression to the base of the hill is 30 degrees.
Visualizing the Scenario
Imagine a right triangle formed by the ship, the base of the hill, and the top of the hill. The ship is at point A, the base of the hill is point B, and the top of the hill is point C. The height of the ship above the water is 10 meters, which we will denote as line segment AD, where D is the point directly below A on the water surface.
Finding the Distance to the Hill
First, let's find the horizontal distance from the ship to the base of the hill (AB). We can use the angle of depression (30 degrees) for this calculation. Since the angle of depression from the ship to the base of the hill is equal to the angle of elevation from the base of the hill to the ship, we can set up the following relationship:
Using the tangent function:
tan(30 degrees) = opposite/adjacent = AD/AB
Here, AD is the height of the ship (10 m), and AB is the distance we want to find.
We know that:
tan(30 degrees) = 1/√3 ≈ 0.577
So, we can write:
0.577 = 10/AB
Rearranging gives us:
AB = 10/0.577 ≈ 17.32 m
Calculating the Height of the Hill
Next, we need to find the height of the hill (BC). We can use the angle of elevation (60 degrees) for this calculation. Again, we will use the tangent function:
tan(60 degrees) = opposite/adjacent = BC/AB
We know that:
tan(60 degrees) = √3 ≈ 1.732
Substituting in the values we have:
1.732 = BC/17.32
Rearranging gives us:
BC = 1.732 * 17.32 ≈ 30 m
Final Results
In summary, the distance from the ship to the base of the hill is approximately 17.32 meters, and the height of the hill is about 30 meters. This example illustrates how trigonometric functions can be applied to real-world scenarios involving angles and distances.
