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The top of a broken tree has its top touching the ground at a distance of 10m from the bottom. If the angle made by the broken part with the ground is 30°, then the length of the broken part is
- a)20m
- b)20√3m
- c)10√3m
- d)20/√3m
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The top of a broken tree has its top touching the ground at a distance...
Let AB be the broken part of the tree.
Therefore, the length of the broken part of the tree is 20/√3 meters.
Most Upvoted Answer
The top of a broken tree has its top touching the ground at a distance...
Understanding the Problem
To find the length of the broken part of the tree, we can visualize the situation as a right triangle. The broken part of the tree forms the hypotenuse, while the distance from the bottom of the tree to the point where the top touches the ground represents one leg of the triangle.
Given Data
- Distance from the base to the point where the tree touches the ground: 10 m
- Angle made by the broken part with the ground: 30 degrees
Using Trigonometric Ratios
In the right triangle formed:
- The distance from the base to the touching point (10 m) is the adjacent side.
- The broken part of the tree (hypotenuse) can be found using the cosine function, which relates the adjacent side and the hypotenuse.
The formula is:
Cosine of angle = Adjacent side / Hypotenuse
Calculation Steps
1. Set up the equation using the cosine function:
- cos(30 degrees) = 10 m / Hypotenuse
2. Knowing that cos(30 degrees) is √3/2, we can substitute:
- √3/2 = 10 / Hypotenuse
3. Rearranging the equation to solve for the hypotenuse:
- Hypotenuse = 10 / (√3/2) = 10 * (2/√3) = 20/√3 m
Conclusion
Therefore, the length of the broken part of the tree is:
- Correct Answer: 20/√3 m (option D)
This approach effectively uses trigonometry to derive the necessary length based on the given angle and distance.
To find the length of the broken part of the tree, we can visualize the situation as a right triangle. The broken part of the tree forms the hypotenuse, while the distance from the bottom of the tree to the point where the top touches the ground represents one leg of the triangle.
Given Data
- Distance from the base to the point where the tree touches the ground: 10 m
- Angle made by the broken part with the ground: 30 degrees
Using Trigonometric Ratios
In the right triangle formed:
- The distance from the base to the touching point (10 m) is the adjacent side.
- The broken part of the tree (hypotenuse) can be found using the cosine function, which relates the adjacent side and the hypotenuse.
The formula is:
Cosine of angle = Adjacent side / Hypotenuse
Calculation Steps
1. Set up the equation using the cosine function:
- cos(30 degrees) = 10 m / Hypotenuse
2. Knowing that cos(30 degrees) is √3/2, we can substitute:
- √3/2 = 10 / Hypotenuse
3. Rearranging the equation to solve for the hypotenuse:
- Hypotenuse = 10 / (√3/2) = 10 * (2/√3) = 20/√3 m
Conclusion
Therefore, the length of the broken part of the tree is:
- Correct Answer: 20/√3 m (option D)
This approach effectively uses trigonometry to derive the necessary length based on the given angle and distance.
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