the top of a building 24meters high is observed from the top and From The Bottom of a vertical tree. the angles of elevation are found to be 45° and 60° respectively find the height of the tree. represent in a diagram first

The diagram shows the situation:

- The tree (height \( h \)) is on the left, and the building (24 meters tall) is on the right.
- From the **top** of the tree, the angle of elevation to the top of the building is \( 45^\circ \).
- From the **bottom** of the tree, the angle of elevation to the top of the building is \( 60^\circ \).

### Using trigonometry:

1. From the top of the tree (with a \( 45^\circ \) angle):
   \[
   \tan(45^\circ) = \frac{\text{height difference}}{\text{distance}} = \frac{24 - h}{d}
   \]
   \[
   1 = \frac{24 - h}{d} \quad \Rightarrow \quad d = 24 - h \quad \text{(Equation 1)}
   \]

2. From the bottom of the tree (with a \( 60^\circ \) angle):
   \[
   \tan(60^\circ) = \frac{24}{d}
   \]
   \[
   \sqrt{3} = \frac{24}{d} \quad \Rightarrow \quad d = \frac{24}{\sqrt{3}} \quad \text{(Equation 2)}
   \]

Now, using Equation 1 and Equation 2 together to solve for \( h \).

Let's solve this system of equations next.The height of the tree is approximately:

\[
h = 24 - 8\sqrt{3} \approx 10.14 \, \text{meters}
\]

The horizontal distance between the tree and the building is:

\[
d = 8\sqrt{3} \approx 13.86 \, \text{meters}
\]

Thus, the tree is about **10.14 meters** tall.

Do you have any questions, or would you like further details? Here are some additional questions you might explore:

1. What would happen if the angle of elevation from the top of the tree was different?
2. How does changing the distance between the tree and the building affect the tree's height?
3. Can we solve this using sine or cosine laws instead of tangent?
4. How would you handle the situation if the building height were unknown but the angles and tree height were given?
5. What if the angles of elevation were swapped—would the result change?

**Tip**: Always check if your calculator is in degrees or radians mode when working with trigonometric functions!

The top of a building 24 meters high is observed from the top and from the bottom of a vertical tree. The angles of elevation are found to be 45° and 60°, respectively. Find the height of the tree. Represent in a diagram first.

Learn how to calculate the height of a tree by applying trigonometry. This problem involves observing angles of elevation from the top and bottom of the tree to a 24-meter building. Suitable for students in Grades 10-12.

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