A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of 30 degrees with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Find the height of the tree.

Step 1: Understand the problem and define variables

We are given a scenario where a tree breaks due to a storm. Let's represent the original tree as a line segment AC. The tree breaks at a point B, and the broken part BC bends to touch the ground at D. The part AB remains vertical.

Step 2: Form a right-angled triangle

The part of the tree that remains standing (AB) is perpendicular to the ground (AD). Therefore, the triangle ABD formed by the standing part of the tree, the ground, and the broken part (BD) is a right-angled triangle at A.

Step 3: Identify given values

We are given that the distance from the foot of the tree (A) to the point where the top touches the ground (D) is 8 m. The angle made by the broken part with the ground is $30^\circ$. Let $h_1$ be the height of the standing part (AB) and $h_2$ be the length of the broken part (BD).

Step 4: Calculate the standing part of the tree (AB)

In $\triangle \text{ABD}$, we use the tangent function to find the height of the standing part (AB), which is opposite to the angle $30^\circ$. We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Substituting the values, we get $h_1 = \frac{8}{\sqrt{3}}$ m.

Step 5: Calculate the broken part of the tree (BD)

To find the length of the broken part (BD), which is the hypotenuse, we use the cosine function. We know $\cos(30^\circ) = \frac{\sqrt{3}}{2}$. Substituting the values, we get $h_2 = \frac{16}{\sqrt{3}}$ m.

Step 6: Find the total height of the tree

The total height of the tree is the sum of the standing part (AB) and the broken part (BC, which is equal to BD). Adding the calculated values for $h_1$ and $h_2$, we get $\frac{24}{\sqrt{3}}$ m.

Step 7: Rationalize the denominator

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{3}$. This simplifies the expression to $\frac{24\sqrt{3}}{3}$, which further simplifies to $8\sqrt{3}$ meters.

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