A person standing on the ground observes the top of a tree at an angle of elevation of 45 degrees. If the person is 10 meters away from the base of the tree, what is the height of the tree?

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UpStudy AI · Accepted Answer

To determine the height of the tree, we can use basic trigonometry involving right-angled triangles.

**Given:**
- **Angle of Elevation ($$ \theta $$)** = 45 degrees
- **Distance from the base of the tree ($$ d $$)** = 10 meters

**We need to find:** 
- **Height of the tree ($$ h $$)**

**Approach:**

1. **Visual Representation:**
   - Imagine a right-angled triangle where:
     - The **adjacent side** is the distance from the person to the tree's base ($$ d = 10 $$ meters).
     - The **opposite side** is the height of the tree ($$ h $$).
     - The **angle of elevation** at the person's eye level is 45 degrees.

2. **Trigonometric Relation:**
   - For a right-angled triangle, the tangent of the angle of elevation relates the opposite side to the adjacent side.
   $$
   \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d}
   $$
   
3. **Calculate the Height:**
   - Plugging in the known values:
   $$
   \tan(45^\circ) = \frac{h}{10}
   $$
   - We know that $$ \tan(45^\circ) = 1 $$, so:
   $$
   1 = \frac{h}{10} \implies h = 10 \text{ meters}
   $$

**Conclusion:**
The height of the tree is **10 meters**.
The tree is 10 meters tall.

A person standing on the ground observes the top of a tree at an angle of elevation of 45 degrees. If the person is 10 meters away from the base of the tree, what is the height of the tree?

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