The angles of elevation of the top of a tree 220 m high from two points in the same vertical plane are 30° and 45°. What is the distance between these two points in metres?

Introduction / Context:
In this problem, two observers are in the same vertical plane with a tall tree. They see the top of the tree under different angles of elevation, 30 degrees and 45 degrees. Given the height of the tree, we must use trigonometry to find how far each observer is from the tree and then determine the distance between the two observers.

Given Data / Assumptions:

• Height of the tree, h = 220 m.
• Angles of elevation from the two points = 30 degrees and 45 degrees.
• The two points lie in the same vertical plane with the tree and on the same side of it.
• The ground is horizontal and the tree is vertical.
• Standard values: tan 30 = 1 / sqrt(3), tan 45 = 1.

Concept / Approach:
Let the distances from the tree base to the two points be d1 and d2, with d1 > d2, corresponding to angles 30 degrees and 45 degrees respectively (smaller angle means farther distance). We use tan θ = height / base to compute d1 and d2. Then the distance between the two points is simply d1 − d2. Finally, using √3 ≈ 1.732, we approximate the numeric value to match one of the options.

Step-by-Step Solution:
Let d1 be the distance to the point where the angle of elevation is 30 degrees. tan 30 = 220 / d1 ⇒ 1 / √3 = 220 / d1 ⇒ d1 = 220√3 m. Let d2 be the distance to the point where the angle of elevation is 45 degrees. tan 45 = 220 / d2 ⇒ 1 = 220 / d2 ⇒ d2 = 220 m. Distance between the two points = d1 − d2 = 220√3 − 220 = 220(√3 − 1) m. Using √3 ≈ 1.732, √3 − 1 ≈ 0.732. Thus distance ≈ 220 × 0.732 ≈ 161.04 m (rounded to 161.05 m).

Verification / Alternative check:
Check d1: 220√3 ≈ 220 × 1.732 ≈ 381 m. Then tan 30 ≈ 220 / 381 ≈ 0.577, matching 1 / √3. Check d2: 220 / 220 = 1, matching tan 45. The difference 381 − 220 ≈ 161 m aligns with the approximate value obtained by the algebraic expression 220(√3 − 1). Everything is consistent with the given angles and height.

Why Other Options Are Wrong:

• 193.22 m and 176.12 m: These overestimate the separation and are not supported by the exact expression 220(√3 − 1).
• 144.04 m and 150.00 m: These underestimate the distance and again do not correspond to 220(√3 − 1).
• Only 161.05 m matches the accurate computation using standard trigonometric values.

Common Pitfalls:
One common error is to add d1 and d2 instead of subtracting them, which would suggest that the points are on opposite sides of the tree. Another mistake is to confuse tan 30 with tan 60 or to misassign the larger distance to the larger angle. Remember that for the same height, a smaller angle of elevation always corresponds to a larger horizontal distance.

Final Answer:
The distance between the two points is approximately 161.05 m.