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

Introduction / Context:
This problem is similar to others involving heights observed from two different locations. Here, the tree height is known, and the angles of elevation from two points are given. We must find the horizontal distance between these observation points by finding each distance from the tree and then subtracting.

Given Data / Assumptions:

• Height of the tree = 220 m.
• Angle of elevation to the top from one point = 30°.
• Angle of elevation to the top from the other point = 45°.
• Both points lie on the same straight line on level ground with the foot of the tree.
• We are asked for the distance between the two observation points.

Concept / Approach:
For a vertical tree of height h and horizontal distance d from a point, we use:
tan(theta) = h / d ⇒ d = h / tan(theta)The point with the smaller angle is farther from the tree. After computing d30 and d45, the distance between the points is |d30 − d45|.

Step-by-Step Solution:
Let h = 220 m.For the 30° observation: tan(30°) = 1 / √3.Thus, distance d30 = h / tan(30°) = 220 / (1 / √3) = 220√3 m.For the 45° observation: tan(45°) = 1.So, distance d45 = h / tan(45°) = 220 / 1 = 220 m.Distance between the two points = d30 − d45 = 220√3 − 220.Factor: = 220(√3 − 1) m.Approximating √3 ≈ 1.732 ⇒ √3 − 1 ≈ 0.732.So, distance ≈ 220 * 0.732 ≈ 161.04 m, usually rounded to 161.05 m.

Verification / Alternative check:
Taking d30 ≈ 220 * 1.732 ≈ 381.04 m, and d45 = 220 m, we get their difference as 161.04 m, consistent with 161.05 m to two decimal places. This confirms our symbolic expression as well as the numerical option provided.

Why Other Options Are Wrong:
193.22 m, 144.04 m, 176.12 m: These are alternate distances that do not correspond to 220(√3 − 1). Substituting them back does not give consistent tangent values.220(√3 - 1) m is the exact symbolic value; numerically it matches approximately 161.05 m, not any of the other decimals listed.

Common Pitfalls:
Some learners mistakenly add the distances instead of subtracting or mix up which angle corresponds to which distance (bigger angle means closer point). Others miscalculate tan(30°) or forget to use the reciprocal when moving from tan(30°) to distance. Using an accurate approximation for √3 is important for numerical answers.

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