Question

A student stand 20m away from the foot of a tree and observe that the angel of elevation of the top of the tree , measured from the table 1.5m above the ground ,is 34°28, calculate the height of the tree to the nearest metre

Answer 1

To calculate the height of the tree, we can follow these steps:

1. **Understand the problem setup**:
- The distance from the student to the foot of the tree is 20 meters.
- The height of the student's eye level (1.5 meters) is taken into account.
- The angle of elevation to the top of the tree from the student's eye level is \(34^\circ 28'\).

2. **Convert the angle from degrees and minutes**:
\[
34^\circ 28' = 34 + \frac{28}{60} = 34.4667^\circ \text{ (approximately)}
\]

3. **Identify the variables**:
- Let \(h\) be the height of the tree.
- The height of the tree above the student's eye level will be \(h - 1.5\) meters.

4. **Use the tangent function**:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In our case:
\[
\tan(34.4667^\circ) = \frac{h - 1.5}{20}
\]

5. **Rearranging the equation**:
\[
h - 1.5 = 20 \cdot \tan(34.4667^\circ)
\]

6. **Calculating \(\tan(34.4667^\circ)\)**:
\[
\tan(34.4667^\circ) \approx 0.687
\]

7. **Substituting this value back into the height equation**:
\[
h - 1.5 = 20 \cdot 0.687 \approx 13.74
\]

8. **Solving for \(h\)**:
\[
h = 13.74 + 1.5 \approx 15.24 \text{ meters}
\]

9. **Rounding to the nearest meter**:
\[
h \approx 15 \text{ meters}
\]

Thus, the height of the tree is approximately **15 meters**.