The base of a tree is situated 50 metres
due north of a point P
. The angle of elevation of the top of the tree from P
is 32°
.

a.
Find the height of the tree, correct to one decimal place.

To solve the problem, we can use trigonometry, specifically the tangent function, because we are dealing with a right triangle where we know one angle and the length of the side adjacent to the angle, and we need to find the length of the side opposite to the angle.

### Given:
- The distance from the point \( P \) to the base of the tree (adjacent side) = 50 meters
- The angle of elevation \( \theta \) = 32°
- We need to find the height of the tree (opposite side).

### Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}
\]
\[
\tan(32°) = \frac{\text{height of the tree}}{50}
\]
\[
\text{height of the tree} = 50 \times \tan(32°)
\]

Now, let's calculate the height.

\[
\text{height of the tree} \approx 50 \times 0.6249 \approx 31.2 \text{ meters}
\]

### Answer:
The height of the tree is approximately **31.2 meters**, correct to one decimal place.

Would you like more details on this calculation or have any questions?

### Follow-up Questions:
1. How would the height change if the angle of elevation were increased to 45°?
2. What if the distance from the point \( P \) to the tree were doubled?
3. Can you find the distance from the top of the tree to the point \( P \)?
4. What would be the effect on the height calculation if the angle of elevation was measured inaccurately?
5. How does changing the angle of elevation affect the tangent function?
6. How can this problem be solved using the sine or cosine functions?
7. What are other real-life applications of using trigonometry in elevation problems?
8. If the ground between point \( P \) and the tree was sloped, how would that affect the calculation?

### Tip:
Always ensure your calculator is set to the correct mode (degrees or radians) based on the angle measurements you are working with.

Learn how to calculate the height of a tree using trigonometry and the angle of elevation. Solve the problem where the base of the tree is 50 meters north of point P, and the angle of elevation to the top of the tree from P is 32°. Follow the step-by-step solution using the tangent function and understand its application in real-life scenarios. Suitable for Grades 9-12.

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