Math Problem Statement
Solution
The problem in the image describes a tree that has broken due to a storm. Here is the summary of the problem:
- The broken part of the tree touches the ground.
- The distance between the base of the tree (its root) and the point where the broken part touches the ground is 8 meters.
- The broken part makes an angle of 60° with the ground.
We are asked to find the total height of the tree.
Solution:
This problem can be solved using trigonometry. The scenario describes a right triangle formed by:
- The vertical part of the tree that remains upright (let's call it ).
- The broken part of the tree (let's call its length ).
- The distance on the ground between the tree base and the point where the tree touches the ground is given as 8 m.
The broken part forms a hypotenuse in this right triangle, where:
- The adjacent side (along the ground) is 8 m.
- The angle formed with the ground is .
Using the cosine function:
Since , we can solve for :
Now that we know the length of the broken part of the tree ( m), we can find the height of the upright part of the tree using the sine function: Since , we get:
Finally, the total height of the tree is the sum of (the upright part) and (the broken part):
Conclusion:
The total height of the tree is approximately 29.86 meters.
Would you like more details or have any questions?
Here are 5 related questions for further exploration:
- How would the total height change if the broken part made a 45° angle with the ground?
- How does the use of trigonometry help in solving real-world problems like this?
- What would be the total height if the ground distance was 10 meters?
- How can the Pythagorean theorem be applied in problems involving right triangles?
- What is the relationship between the angle and the length of the broken part of the tree?
Tip: Always sketch a diagram when solving geometry or trigonometry problems to help visualize the relationships between angles and sides.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangles
Formulas
cos(θ) = adjacent / hypotenuse
sin(θ) = opposite / hypotenuse
Theorems
Trigonometric Ratios
Suitable Grade Level
Grades 9-12
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