From the top of a building 10m high, the angle of elevation of a fruit on top of a tree 25m is 30º. Calculate…

Analysis

To solve this problem, we need to visualize the geometry of the situation. We have a building and a tree standing on horizontal ground. We are looking for the horizontal distance between them.

Concept: Trigonometry (Right-Angled Triangles)

When dealing with angles of elevation, we use the tangent ratio:

In this scenario:

• The height of the building is $10\text{m}$.
• The height of the tree is $25\text{m}$.
• The angle of elevation from the top of the building to the top of the tree is $30^\circ$.

Step-by-Step Solution

1. Find the vertical difference: Since the observer is at the top of the $10\text{m}$ building looking at the top of the $25\text{m}$ tree, the "opposite" side of our triangle is the difference in their heights: $$\text{Opposite} = 25\text{m} - 10\text{m} = 15\text{m}$$

2. Identify the variables: • $\text{Opposite} = 15\text{m}$ • $\text{Angle} (\theta) = 30^\circ$ • $\text{Adjacent} (\text{horizontal distance}) = x$

3. Apply the formula: $$\tan(30^\circ) = \frac{15}{x}$$

4. Solve for $x$: We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. $$\frac{1}{\sqrt{3}} = \frac{15}{x}$$ $$x = 15 \times \sqrt{3}$$ $$x = 15\sqrt{3}\text{m}$$

Conclusion

The horizontal distance between the building and the tree is $15\sqrt{3}\text{m}$. This matches option A.

Exam Tip: Always remember to subtract the height of the observer (the building) from the height of the object (the tree) when calculating angles of elevation from a point above the ground. If you don't subtract, you will get the wrong "opposite" side!