A palm tree 90 ft high, is broken by the wind and its upper part meet the ground at an angle of `30^(@)`. Find the distance of the point where the top of the tree meets the ground from its root:

Options

A. 43.69 ft
B. 51.96 ft
C. 60 ft
D. 30 ft

Answer

To solve the problem, we need to find the distance from the root of the palm tree to the point where the top of the tree meets the ground after being broken. Let's break down the solution step by step.

### Step 1: Understand the Geometry
We have a palm tree that is originally 90 ft high. When it breaks, the top part of the tree falls and makes contact with the ground at an angle of 30 degrees. We can visualize this as a right triangle where:
- The height of the tree (90 ft) is one leg of the triangle.
- The distance from the base of the tree to the point where the top meets the ground is the other leg of the triangle.
- The broken part of the tree acts as the hypotenuse.

### Step 2: Define Variables
Let:
- \( AC \) = height of the tree that remains standing (unknown).
- \( CD \) = length of the broken part of the tree (hypotenuse).
- \( AD \) = distance from the base of the tree to where the top meets the ground (unknown).
- Since the total height of the tree is 90 ft, we can write:
\[ AC + CD = 90 \text{ ft} \]

### Step 3: Use Trigonometric Ratios
From the triangle formed, we can use the sine function to relate the angle and the sides:
- The angle \( \angle CDB = 30^\circ \).
- The opposite side to this angle is \( AC \) (which is unknown) and the hypotenuse is \( CD \).

Using the sine function:
\[ \sin(30^\circ) = \frac{AC}{CD} \]
We know that \( \sin(30^\circ) = \frac{1}{2} \), so we can write:
\[ \frac{1}{2} = \frac{AC}{CD} \]

### Step 4: Express \( AC \) in Terms of \( CD \)
Rearranging gives:
\[ AC = \frac{1}{2} CD \]

### Step 5: Substitute into the Height Equation
Substituting \( AC \) into the height equation:
\[ \frac{1}{2} CD + CD = 90 \]
This simplifies to:
\[ \frac{3}{2} CD = 90 \]
Thus,
\[ CD = 90 \times \frac{2}{3} = 60 \text{ ft} \]

### Step 6: Find \( AC \)
Now substituting back to find \( AC \):
\[ AC = \frac{1}{2} \times 60 = 30 \text{ ft} \]

### Step 7: Find the Distance \( AD \)
Now we can use the cosine function to find the distance \( AD \):
\[ \cos(30^\circ) = \frac{AD}{CD} \]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so:
\[ \frac{\sqrt{3}}{2} = \frac{AD}{60} \]
Rearranging gives:
\[ AD = 60 \times \frac{\sqrt{3}}{2} = 30\sqrt{3} \text{ ft} \]

### Step 8: Calculate the Numerical Value
Now, calculating \( 30\sqrt{3} \):
\[ 30 \times 1.732 = 51.96 \text{ ft} \]

### Final Answer
The distance from the point where the top of the tree meets the ground to its root is approximately **51.96 ft**.

---