a rope dancer was walking on a loose rope 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.
Unless you give the problem a different interpretation, Arora gave you a solution back here:
https://www.jiskha.com/display.cgi?id=1516706057
Did you look at it? What don't you agree with?
It said" a loose rope". By that I understood, as did Arora, that the rope is not straight across but sags.
To solve this problem, we can use trigonometry. Let's break it down into steps:
Step 1: Visualize the problem.
Draw a diagram of the situation to help understand the question. You have two equal posts of height 9m, and a rope dancer walking on a loose rope tied between them. The rope dancer is 3m above the ground.
A
/ \
/ \
/ \
/-------\
/| |\
/ | | \
/ | | \
/ | | \
B---|-------|---C
9m 9m
We have three points: A - where the rope is tied above the left post, B - where the rope dancer is positioned, and C - where the rope is tied above the right post.
Step 2: Identify the given information.
From the problem, we know that the height of the posts is 9m, and the rope dancer is 3m above the ground. We also know that the stretched pieces of rope make angles of 30 degrees and 60 degrees with the horizontal line parallel to the ground.
Step 3: Break down the problem.
Let's split the problem into two parts:
1. Find the horizontal distance between the two posts, which will give us the length of the rope.
2. Use trigonometry to find the length of the rope.
Step 4: Find the horizontal distance between the two posts.
To find the horizontal distance, we need to use the information about the angles made by the stretched pieces of rope.
In triangle ABC, the angle ADC is a right angle because it is perpendicular to the ground. Therefore, angle ADB and angle BDC are complementary. Given that angle ADB is 60 degrees, angle BDC must be 30 degrees.
We are given that the height of the posts is 9m and the rope dancer is 3m above the ground. Therefore, the vertical distance between point B (where the rope dancer is) and the horizontal line parallel to the ground is 9m - 3m = 6m.
Using trigonometry, we can find the horizontal distance between the two posts:
tan(30 degrees) = horizontal distance / 6m
horizontal distance = 6m * tan(30 degrees)
horizontal distance = 3.464m
So, the horizontal distance between the two posts is approximately 3.464m.
Step 5: Use trigonometry to find the length of the rope.
Now that we have the horizontal distance, we can use it to find the length of the rope.
In triangle ABC, the length of the rope is the hypotenuse of a right triangle. The vertical distance (from the ground to the rope dancer) is 3m, and the horizontal distance is 3.464m.
Using the Pythagorean theorem, we can find the length of the rope:
Length of rope^2 = (Vertical distance)^2 + (Horizontal distance)^2
Length of rope^2 = 3m^2 + 3.464m^2
Length of rope^2 = 9m^2 + 11.981m^2
Length of rope^2 = 20.981m^2
Length of rope = √(20.981m^2)
Length of rope ≈ 4.58m
Therefore, the length of the rope is approximately 4.58m.