This problem involves calculating the height of a tree using trigonometry, specifically the concept of the angle of elevation. When we look at the top of a tree from a point on the ground, the angle formed between our line of sight and the horizontal ground is called the angle of elevation. This scenario typically forms a right-angled triangle, where the tree's height is one side, the distance from the base of the tree to the observation point is another side, and the line of sight is the hypotenuse.

Tree Height Calculation Overview

To find the height of the tree, we can model the situation as a right-angled triangle. Let's define the parts of this triangle:

• Let 'h' be the height of the tree (the side opposite to the angle of elevation).
• Let 'd' be the distance from the base of the tree to the point on the level ground (the side adjacent to the angle of elevation). This distance is given as 15 m.
• The angle of elevation from the point on the level ground to the top of the tree is given as 45°.

Trigonometric Ratios Application

In a right-angled triangle, the trigonometric ratio that relates the opposite side (height of the tree) to the adjacent side (distance from the base) is the tangent function. The formula for tangent is:

$$\text{tan}(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$$

In our specific problem:

• The angle, \(\theta\), is 45°.
• The Opposite Side is the height of the tree, which we want to find (h).
• The Adjacent Side is the distance from the base of the tree to the point on the ground, which is 15 m.

So, we can set up the equation as:

$$\text{tan}(45^\circ) = \frac{h}{15}$$

Height Calculation Steps

We know a standard trigonometric value for \(\text{tan}(45^\circ)\). The value of \(\text{tan}(45^\circ)\) is 1. Now, we substitute this value into our equation:

$$1 = \frac{h}{15}$$

To solve for 'h', we need to isolate it. We can do this by multiplying both sides of the equation by 15:

$$h = 1 \times 15$$

$$h = 15 \text{ m}$$

Therefore, the height of the tree is 15 meters.

Here's a summary of the given information and the calculated result:

This result demonstrates that when the angle of elevation from a point on the ground to the top of an object (like a tree) is 45°, the height of the tree is exactly equal to the horizontal distance from the observation point to its base. This is a unique property of a right-angled isosceles triangle, where the two perpendicular sides (the height and the base distance) are equal in length.