A flagstaff 17.5 m high casts a shadow 40.25 m long. Under the same conditions, a building casts a shadow 28.75 m long. What is the height of the building?

Introduction:
This question is a classic application of similar triangles in height-and-distance problems. The heights of vertical objects and lengths of their shadows are proportional when the angle of elevation of the light source remains the same, such as under constant sunlight.

Given Data / Assumptions:

• Height of the flagstaff = 17.5 m.
• Length of the flagstaff's shadow = 40.25 m.
• Length of the building's shadow = 28.75 m.
• The flagstaff and the building are vertical and exposed to the same light source at the same time.
• We must determine the height of the building.

Concept / Approach:
When two vertical objects cast shadows under identical lighting, triangles formed by each object and its shadow are similar. This implies:
Height of flagstaff / Shadow of flagstaff = Height of building / Shadow of buildingWe use this proportion to compute the unknown height.

Step-by-Step Solution:
Step 1: Set up the proportion.17.5 / 40.25 = Height of building / 28.75Step 2: Express height of building H.H = 28.75 * (17.5 / 40.25)Step 3: Compute numerically.First, notice 17.5 / 40.25 is a constant ratio.Using direct calculation, H = (28.75 * 17.5) / 40.25This evaluates to H = 12.5 m

Verification / Alternative check:
Compare the ratios. For the flagstaff, height : shadow = 17.5 : 40.25. For the building, using H = 12.5 m and shadow 28.75 m, ratio = 12.5 : 28.75. Both ratios simplify to the same value numerically, confirming similarity of triangles and the correctness of H = 12.5 m.

Why Other Options Are Wrong:
14 m, 13.5 m, 11.4 m, 16 m: Substituting any of these values into the proportion H / 28.75 = 17.5 / 40.25 yields a mismatch; the ratios of height to shadow would not be equal to that of the flagstaff. Only 12.5 m preserves the proportional relationship required by similar triangles.

Common Pitfalls:
Typical mistakes include using the inverse ratio (shadow/height) incorrectly or not maintaining consistent units. Another error is rough mental approximation without checking that both ratios are equal. Writing the proportion carefully and performing stepwise calculation prevents such errors.

Final Answer:
The height of the building is 12.5 m.