In a regular hexagon ABCDEF, two vertical towers stand at vertices B and C. From vertex A on the ground, the angle of elevation to the top of the tower at B is 30°, and the angle of elevation to the top of the tower at C is 45°. Assuming all vertices lie on the same horizontal plane, what is the ratio of the heights of the towers at B and C (h_B : h_C)?

Introduction / Context:
This is a geometry and trigonometry hybrid problem involving a regular hexagon and angles of elevation. By using properties of a regular hexagon and basic tangent definitions, we can relate the horizontal distances from A to B and from A to C with the heights of the towers placed at those vertices. Such questions appear in height and distance sections of aptitude exams, often combined with polygon properties.

Given Data / Assumptions:

• ABCDEF is a regular hexagon.
• Vertical towers of heights h_B and h_C stand at vertices B and C respectively.
• From vertex A, angle of elevation to the top at B is 30°.
• From vertex A, angle of elevation to the top at C is 45°.
• All vertices of the hexagon lie on the same horizontal plane.
• We need the ratio h_B : h_C.

Concept / Approach:
In a regular hexagon of side length s, each side is equal, and all interior angles are 120°. If the hexagon is inscribed in a circle, the central angle between adjacent vertices is 60°. The distances AB and AC can be expressed in terms of the side length s. Then, using tan(angle) = height / horizontal distance, we can write h_B and h_C in terms of s and the given angles. Finally, dividing h_B by h_C eliminates s and gives the required ratio.

Step-by-Step Solution:
Let the side length of the regular hexagon be s. Then AB = s, because adjacent vertices are one side apart. Vertices of a regular hexagon on a circle are 60° apart at the center. The distance AC corresponds to a chord that spans two sides, that is, 120° at the center. For a circle of radius R, chord length for 60° is AB = 2R sin 30° = R, so s = R. Chord length for 120° is AC = 2R sin 60° = 2R * (sqrt(3) / 2) = R sqrt(3) = s sqrt(3). Now use tangents for the right triangles formed with vertical heights. tan 30° = h_B / AB = h_B / s. So h_B = s * tan 30° = s * (1 / sqrt(3)) = s / sqrt(3). tan 45° = h_C / AC = h_C / (s sqrt(3)). Since tan 45° = 1, we have h_C = s sqrt(3). Therefore, h_B : h_C = (s / sqrt(3)) : (s sqrt(3)). Cancel s from both terms: h_B : h_C = 1 / sqrt(3) : sqrt(3). Multiply numerator and denominator by sqrt(3) to simplify: 1 : 3.

Verification / Alternative check:
You can assign a convenient value to s to check the ratio numerically. For example, let s = sqrt(3). Then AB = sqrt(3) and AC = 3. From A, h_B = AB * tan 30° = sqrt(3) * (1 / sqrt(3)) = 1. Also, h_C = AC * tan 45° = 3 * 1 = 3. Thus h_B : h_C = 1 : 3, matching the algebraic result.

Why Other Options Are Wrong:
1 : sqrt(3) would be the ratio if h_C were s rather than s sqrt(3). 1 : 2 and 1 : 2sqrt(3) do not arise from the geometry and correspond to incorrect distance assumptions. sqrt(3) : 1 is the inverse of the correct ratio and would imply h_B is taller than h_C, which contradicts the angles of elevation (45° for C should correspond to a relatively taller or closer tower than 30° for the same horizontal scale).

Common Pitfalls:
A frequent error is to assume AC = 2s instead of s sqrt(3). Another mistake is to forget that the angle of elevation is measured from A to the top of each tower, not from the center of the hexagon. Using wrong chord lengths or misapplying tangent formulas will quickly lead to incorrect ratios. Drawing a rough diagram often helps to visualise the relationships.

Final Answer:
The ratio of the heights of the towers at B and C is 1 : 3.