Understanding the Angles of Elevation Problem

This problem involves calculating the height of a tree using angles of elevation from two different points on the ground. We are given the distance between these two points and the angles of elevation from each point to the top of the tree. The final goal is to find 40% of the tree's height. We need to use trigonometry to solve this.

Key terms: angles of elevation, tree height, distance, trigonometry.

Step-by-Step Calculation of Tree Height

Let's define the variables first:

• Let $h$ be the height of the tree.
• Let the two points on the ground be A and B. They are 50 meters apart.
• Assume point B is closer to the tree than point A.
• Let $x$ be the distance from point B to the base of the tree.
• Since A is 50 meters farther from B on the same side, the distance from point A to the base of the tree is $(x + 50)$ meters.

We are given the angles of elevation:

• Angle of elevation from point B to the top of the tree is $60^\circ$.
• Angle of elevation from point A to the top of the tree is $30^\circ$.

Now, we can set up trigonometric relationships using the tangent function (tangent = opposite/adjacent):

1. From point B: The angle is $60^\circ$, the opposite side is the tree height $h$, and the adjacent side is the distance $x$. So, $\tan(60^\circ) = \frac{h}{x}$. We know that $\tan(60^\circ) = \sqrt{3}$. Therefore, $\sqrt{3} = \frac{h}{x}$, which gives us $h = x\sqrt{3}$.
2. From point A: The angle is $30^\circ$, the opposite side is the tree height $h$, and the adjacent side is the distance $(x + 50)$. So, $\tan(30^\circ) = \frac{h}{x + 50}$. We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Therefore, $\frac{1}{\sqrt{3}} = \frac{h}{x + 50}$, which gives us $h = \frac{x + 50}{\sqrt{3}}$.

Since both expressions equal $h$, we can set them equal to each other to find $x$:

$$ x\sqrt{3} = \frac{x + 50}{\sqrt{3}} $$

Multiply both sides by $\sqrt{3}$:

$$ x\sqrt{3} \times \sqrt{3} = x + 50 $$ $$ 3x = x + 50 $$

Now, solve for $x$:

$$ 3x - x = 50 $$ $$ 2x = 50 $$ $$ x = \frac{50}{2} $$ $$ x = 25 \text{ meters} $$

Now that we have the value of $x$, we can find the height of the tree, $h$. Using the first equation $h = x\sqrt{3}$:

$$ h = 25\sqrt{3} \text{ meters} $$

Calculating 40% of the Tree's Height

The question asks for 40% of the calculated height $h$.

40% of $h = \frac{40}{100} \times h$

Substitute the value of $h$:

$$ \frac{40}{100} \times 25\sqrt{3} $$

Simplify the calculation:

$$ \frac{2}{5} \times 25\sqrt{3} $$

Multiply:

$$ 2 \times \frac{25}{5} \times \sqrt{3} $$ $$ 2 \times 5 \times \sqrt{3} $$ $$ 10\sqrt{3} \text{ meters} $$

So, 40% of the height of the tree is $10\sqrt{3}$ meters.