This problem involves finding the total height of a tree that was broken by a cyclone. We can model this situation using a right-angled triangle where trigonometry is applied.

Problem Setup Visualization

Visualize the scenario:

• A tree stands vertically, then breaks.
• The broken part bends over and touches the ground, forming the hypotenuse of a right-angled triangle.
• The standing part of the tree forms the vertical leg (opposite side).
• The distance along the ground from the root to where the top touches forms the horizontal leg (adjacent side).

Given information:

• The angle between the broken part (hypotenuse) and the ground (adjacent side) is $30^\circ$.
• The distance from the root to the tip touching the ground (adjacent side) is $d = 10$ m.
• We need to find the original height of the tree, which is the sum of the standing part and the broken part.

Trigonometric Calculations

Let $h_{standing}$ represent the height of the standing part of the tree (opposite side).

Let $h_{broken}$ represent the length of the broken part of the tree (hypotenuse).

The angle with the ground is $\theta = 30^\circ$. The adjacent side is $d = 10$ m.

Using trigonometric ratios:

• To find $h_{standing}$ (opposite side): Use the tangent function.

  $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h_{standing}}{d}$

  $h_{standing} = d \times \tan(\theta)$

• To find $h_{broken}$ (hypotenuse): Use the cosine function.

  $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{d}{h_{broken}}$

  $h_{broken} = \frac{d}{\cos(\theta)}$

Calculating Tree Height Components

1. Calculate the standing part ($h_{standing}$):

Substitute the values:

$h_{standing} = 10 \times \tan(30^\circ)$

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.

$h_{standing} = 10 \times \frac{1}{\sqrt{3}} = \frac{10}{\sqrt{3}}$ m.

2. Calculate the broken part ($h_{broken}$):

Substitute the values:

$h_{broken} = \frac{10}{\cos(30^\circ)}$

We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.

$h_{broken} = \frac{10}{\frac{\sqrt{3}}{2}} = 10 \times \frac{2}{\sqrt{3}} = \frac{20}{\sqrt{3}}$ m.

Determining Total Tree Height

The original height of the tree ($H$) is the sum of the standing part and the broken part:

$H = h_{standing} + h_{broken}$

$H = \frac{10}{\sqrt{3}} + \frac{20}{\sqrt{3}}$

$H = \frac{10 + 20}{\sqrt{3}} = \frac{30}{\sqrt{3}}$ m.

Rationalize the denominator to simplify:

$H = \frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{30\sqrt{3}}{3}$

$H = 10\sqrt{3}$ m.

The height of the tree is $10\sqrt{3}$ m.