Two ranging rods, one of 3.00 m and the other of 1.50 m length, were used in the effort to find the height of an in accessible tower. In the first setting the rods were so placed that their tops were inline with the top of the tower. The distance between the rods was 15 m .In the second setting the rods were ranged on the same line as before. This time the distance between the rods was 30 m. If the distance between the two longer rods was 90 m, find the height of the tower

Problem Statement

Two ranging rods, one 3.00 m long and the other 1.50 m long, were used to determine the height of an inaccessible tower. In the first setup, the rods were placed 15 m apart such that their tops aligned with the tower’s top. In the second setup, along the same line, the rods were placed 30 m apart, again with their tops aligned with the tower’s top. The distance between the two positions of the longer (3.00 m) rod was measured to be 90 m. Find the height of the tower.

Diagram illustrating the use of ranging rods at two different settings to determine the height of an inaccessible tower using similar triangles

Step-by-Step Solution

Key Information & Setup

Rod 1 Height = 3.00 m.
Rod 2 Height = 1.50 m.
Setting 1 (Closer to tower): 3m rod at B, 1.5m rod at E. Distance BE = 15 m.
Setting 2 (Further from tower): 3m rod at A, 1.5m rod at C. Distance AC = 30 m.
Distance between 3m rod positions: AB = 90 m.
Points A, C, B, E, F (Tower Base) are collinear in that order.
Let EF = x (Distance from 1.5m rod in Setting 1 to Tower Base F).
Let H be the total height of the tower (HF).
Let FT be the height of the tower *above the 1.5m level* (FT = H – 1.5 m).
Goal: Find H.

Derived Distances:
CB = AB – AC = 90 – 30 = 60 m.
BF = BE + EF = 15 + x.
CF = CB + BF = 60 + (15 + x) = 75 + x.
AF = AC + CF = 30 + (75 + x) = 105 + x.

Step 1: Calculate Tangents (Slopes) of Lines of Sight

The tangent of the angle of elevation (slope) of the line of sight relative to the horizontal line from the top of the shorter (1.5m) rod can be found using the height difference and distance between the rods in each setting.

Setting 1 (Rods at B and E): Let the angle be β.

tan(β) = (Height Rod 1 – Height Rod 2) / Distance BE

tan(β) = (3.00 m – 1.50 m) / 15 m = 1.50 / 15

tan(β) = 0.10

Setting 2 (Rods at A and C): Let the angle be α.

tan(α) = (Height Rod 1 – Height Rod 2) / Distance AC

tan(α) = (3.00 m – 1.50 m) / 30 m = 1.50 / 30

tan(α) = 0.05

Step 2: Set Up Equations for Height Difference (FT)

Let FT be the height of the tower above the 1.5m level (FT = H – 1.5). Based on the geometry of the two setups and the calculated tangents (slopes), we can establish relationships involving FT and the unknown distance x = EF.

The relationship derived from Setting 1 (using tan β) is:

Eq 1: FT = (x + 15) * 0.10

The relationship derived from Setting 2 (using tan α) is:

Eq 2: FT = (x + 105) * 0.05

(Here, x+15 corresponds to distance BF, and x+105 corresponds to distance AF).

Step 3: Solve for x (Distance EF)

Equate the two expressions for FT found in Step 2:

(x + 15) * 0.10 = (x + 105) * 0.05

Multiply both sides by 20 to clear decimals:

2 * (x + 15) = 1 * (x + 105)

2x + 30 = x + 105

2x – x = 105 – 30

x = 75 m

So, the distance from the inner 1.5m rod (at E) to the tower base (F) is 75 m.

Step 4: Calculate Height Difference (FT)

Substitute the value x = 75 m back into either equation for FT (Eq 1 is simpler):

FT = (x + 15) * 0.10

FT = (75 + 15) * 0.10

FT = 90 * 0.10

FT = 9.0 m

This is the height of the tower above the 1.50 m level.

Step 5: Calculate Total Tower Height (H)

The total height of the tower (H) is the height of the shorter rod plus the calculated height difference FT.

H = Height of Rod 2 + FT

H = 1.50 m + 9.0 m

H = 10.5 m

Final Result

The height of the inaccessible tower is 10.5 metres.

Conceptual Explanation & Applications

Core Concepts:

Indirect Measurement: Determining a dimension (height) without direct measurement, often used when the object is inaccessible.
Ranging Rods: Simple surveying tools of known length used for marking points and sighting lines.
Line of Sight Alignment: Ensuring the tops of the ranging rods and the target point (tower top) lie on a single straight line.
Similar Triangles / Slopes: Implicitly, the method relies on the principle that the slope (tangent of the elevation angle) of the line of sight is constant. The height difference between rods divided by their separation distance gives this slope relative to the horizontal.
Simultaneous Equations: Using two different setups (pairs of rod positions) creates two independent geometric constraints, allowing the formation of two equations that can be solved simultaneously for unknowns (like distance ‘x’ and height difference ‘FT’).

Real-World Applications:

Estimating heights of tall structures (towers, chimneys, buildings) where the base cannot be reached or where precise instruments like theodolites are unavailable.
Measuring heights of natural features like tall trees or cliffs.
Educational demonstrations of surveying principles and trigonometry/geometry.
Preliminary or reconnaissance surveys.

Why It Works:
The method leverages the properties of similar triangles formed by the lines of sight. By establishing two distinct lines of sight towards the tower top, each defined by aligning the tops of the two ranging rods (with known height difference) at a known separation, we obtain two different geometric setups. The slope (or tangent of the elevation angle) relative to the horizontal line from the shorter rod’s top is calculated for each setup. This slope relates the horizontal distance to the vertical height difference. Since both lines converge at the tower top, we can create two equations involving the unknown horizontal distance (x = EF) and the unknown height of the tower above the shorter rod (FT = H – 1.5). Solving these two equations simultaneously allows us to find both ‘x’ and ‘FT’. Finally, adding the height of the shorter rod (1.5 m) to FT gives the total tower height H.