a rope dancer was walking on a loose roop tied to the top of two equal post of height 9m. when he was 3m above the ground, it was found that stretched pieces of roop made angle of 30 degree and 60 degree with the horizontal line parallel to the ground. find the length of the rope
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To find the length of the rope, we can use trigonometry. Let's start by visualizing the situation:
1. Draw a diagram representing the two posts of equal height, which we'll label as A and B. Connect them with a horizontal line parallel to the ground to represent the loose rope.
A ____ B
|------|
2. Label the height of the posts as 9m.
A ____ B
|------|
9m
3. The rope dancer is 3m above the ground, so we'll create a triangle ABC, where A represents the ground, B represents the rope dancer, and C represents the top of the post.
A ____ B
|-------|
9m
|-----|
3m
Now, we need to find the length of the rope, which is represented by the line segment BC.
To find BC, we need to use the trigonometric ratios sine and cosine.
In the triangle ABC, the angles CAB and CBA are given as 30 degrees and 60 degrees, respectively.
Using the given information, we know that the angle CBA is complementary to the angle CAB because they add up to 90 degrees.
Now, let's solve for BC using trigonometry:
1. We can start by finding the length of the segment AC using sine:
sin(30°) = opposite / hypotenuse
sin(30°) = 3m / AC
Rearranging the equation, we get:
AC = 3m / sin(30°)
2. Next, we can find the length of the segment AB using cosine:
cos(30°) = adjacent / hypotenuse
cos(30°) = AB / AC
Rearranging the equation, we get:
AB = AC * cos(30°)
3. Finally, we can find the length of the segment BC using cosine again:
cos(60°) = adjacent / hypotenuse
cos(60°) = BC / AB
Rearranging the equation, we get:
BC = AB * cos(60°)
Now, let's substitute the values we found:
AC = 3m / sin(30°)
AB = AC * cos(30°)
BC = AB * cos(60°)
Calculating these equations, we can find the length of the rope.