from the top of a vertical mast 150m high,two huts on the same ground level are observed. One due east and the other due west of the mast. Their angles of depression are 60° and 45° respectively. Find the distance
I assume you want the distance between the huts. If you draw a diagram and review your basic trig functions, it should be clear that the distance is
150 cot60° + 150 cot45°
from the top of a vertical mast 150m high,two huts on the same ground level are observed. One due east and the other due west of the mast. Their angles of depression are 60° and 45° respectively. Find the distance
nice repost. so what now?
To find the distance between the mast and the huts, we can use trigonometry and the concept of angles of depression.
Let's start by visualizing the scenario. We have a vertical mast that is 150 meters high. From the top of the mast, we observe two huts on the ground level, one due east and the other due west of the mast. The angles of depression to the huts are given as 60° and 45° respectively.
Now, let's name the distance we want to find as "x" as shown in the diagram below:
Hut (60°)
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/
/
/
/
Mast (150m)---------------- Ground ---------------- Hut (45°)
To find the distance, we will focus on the right triangle formed by the mast, the ground, and the hut with a 60° angle of depression.
In a right triangle, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the tangent of the 60° angle can be written as:
tan(60°) = opposite side / adjacent side
We can rewrite this equation as:
tan(60°) = 150 / x
To solve for x, we can rearrange the equation as follows:
x = 150 / tan(60°)
Using a calculator, we can find that the tangent of 60° is approximately 1.732. Therefore:
x = 150 / 1.732
x ≈ 86.60 meters
So the distance between the mast and the hut to the east is approximately 86.60 meters.
Similarly, we can find the distance between the mast and the hut to the west by focusing on the right triangle formed by the mast, the ground, and the hut with a 45° angle of depression.
Applying the same logic, we get:
tan(45°) = opposite side / adjacent side
tan(45°) = 150 / x
x = 150 / tan(45°)
Using a calculator, we can find that the tangent of 45° is exactly 1. Therefore:
x = 150 / 1
x = 150 meters
So the distance between the mast and the hut to the west is 150 meters.
In summary, the distance between the mast and the hut to the east is approximately 86.60 meters, and the distance between the mast and the hut to the west is 150 meters.