A tree of height 15 m is broken by wind in such a way that its top touches the ground and makes an angle 30° with the ground. What is the height from the ground to the point where tree is broken?
5 m
Understanding the Problem: Broken Tree and Trigonometry
This problem involves a tree that breaks due to wind. The tree forms a right-angled triangle when its broken top touches the ground. We are given the total height of the tree and the angle the broken part makes with the ground. We need to find the height from the ground where the tree broke.
Setting up the Geometry
Let's visualize the situation:
- The original tree stood vertically.
- When it broke, the upper part bent over, with the break point acting as a pivot.
- The part of the tree still standing forms one side of a right triangle (the vertical leg).
- The broken part of the tree forms the hypotenuse of the right triangle.
- The ground from the base of the tree to where the top touches forms the other side (the horizontal leg).
Let:
- \( H \) be the total height of the tree = 15 m.
- \( h \) be the height from the ground where the tree broke (the standing part). This is what we need to find.
- \( x \) be the length of the broken part of the tree.
Since the broken part \( x \) and the standing part \( h \) make up the total height of the tree, we have the relationship:
\[ H = h + x \] \[ 15 = h + x \]From this, we can express \( x \) in terms of \( h \):
\[ x = 15 - h \]Applying Trigonometry to the Broken Tree
The problem states that the top of the broken tree touches the ground and makes an angle of 30° with the ground. In the right-angled triangle formed:
- The angle is 30°.
- The side opposite to the 30° angle is the standing part of the tree, which is \( h \).
- The hypotenuse is the broken part of the tree, which is \( x \) (or \( 15 - h \)).
The trigonometric function that relates the opposite side and the hypotenuse is the sine function:
\[ \sin(\text{angle}) = \frac{\text{Opposite side}}{\text{Hypotenuse}} \]Plugging in the values and variables for our broken tree problem:
\[ \sin(30^\circ) = \frac{h}{x} \]We know that \( \sin(30^\circ) = \frac{1}{2} \) and we have \( x = 15 - h \). Substituting these into the equation:
\[ \frac{1}{2} = \frac{h}{15 - h} \]Solving for the Broken Height
Now, we solve this equation for \( h \):
- Cross-multiply: \[ 1 \times (15 - h) = 2 \times h \] \[ 15 - h = 2h \]
- Add \( h \) to both sides of the equation: \[ 15 = 2h + h \] \[ 15 = 3h \]
- Divide both sides by 3: \[ h = \frac{15}{3} \] \[ h = 5 \]
So, the height from the ground to the point where the tree broke is 5 meters.
Summary of Calculation
| Parameter | Value/Expression |
|---|---|
| Total tree height (H) | 15 m |
| Height where tree broke (h) | ? |
| Length of broken part (x) | \(15 - h\) |
| Angle with ground | 30° |
| Trigonometric relation | \( \sin(30^\circ) = \frac{h}{x} \) |
| Equation | \( \frac{1}{2} = \frac{h}{15 - h} \) |
| Calculated broken height (h) | 5 m |
The height from the ground to the point where the tree is broken is 5 meters.
Revision Table: Broken Tree Trigonometry
| Concept | Description |
|---|---|
| Right Triangle | A triangle with one angle measuring 90 degrees. Used here to model the tree, ground, and broken part. |
| Hypotenuse | The side opposite the right angle in a right triangle. In this case, the broken part of the tree. |
| Opposite Side | The side across from a given angle in a right triangle. Here, the standing part of the tree is opposite the 30° angle. |
| Sine Function (\( \sin \)) | In a right triangle, the ratio of the length of the opposite side to the length of the hypotenuse (\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)). Essential for solving this problem. |
| Angle of Elevation/Depression | Angle measured from the horizontal. Here, the angle the broken part makes with the ground is an angle related to elevation. |
Additional Information: Solving Trigonometry Problems
Problems involving heights, distances, and angles can often be solved using trigonometry. Here are some key points:
- Always draw a diagram to visualize the problem. This helps identify the right triangle and the known/unknown sides and angles.
- Label the sides of the right triangle relative to the given angle: opposite, adjacent, and hypotenuse.
- Choose the appropriate trigonometric function (sine, cosine, or tangent) based on which sides and angles are involved (SOH CAH TOA mnemonic can be helpful).
SOH: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
CAH: \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
TOA: \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)
- Use known trigonometric values for standard angles (like 0°, 30°, 45°, 60°, 90°) or a calculator for others.
- Set up the equation and solve for the unknown variable.
This broken tree problem is a classic example demonstrating the application of the sine function in a real-world scenario involving heights and angles.
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