A vertical pole of 28 m height casts a 19.2 m long shadow. At the same time, find the length of the shadow cast by another pole of 52.5 m height.
36 m
Calculating Shadow Length Using Proportion and Similar Triangles
When the sun is at a particular angle, any vertical object and its shadow form a right-angled triangle. At the same time, the angle of elevation of the sun is the same for all objects in the vicinity. This means that the right-angled triangles formed by different vertical poles and their shadows are similar triangles.
In similar triangles, the ratio of corresponding sides is equal. In this case, the ratio of the height of the pole to the length of its shadow is constant.
Setting up the Proportion for Pole Height and Shadow Length
Let:
H1be the height of the first pole = 28 mS1be the length of the shadow of the first pole = 19.2 mH2be the height of the second pole = 52.5 mS2be the length of the shadow of the second pole (unknown)
Since the triangles are similar, the ratio of height to shadow length is the same for both poles:
\[\frac{H_1}{S_1} = \frac{H_2}{S_2}\]
Substitute the given values into the equation:
\[\frac{28}{19.2} = \frac{52.5}{S_2}\]
Solving for the Unknown Shadow Length
To find the length of the shadow cast by the second pole (\(S_2\)), we can cross-multiply and solve the equation:
\[28 \times S_2 = 19.2 \times 52.5\]
Calculate the product on the right side:
\[19.2 \times 52.5 = 1008\]
So, the equation becomes:
\[28 \times S_2 = 1008\]
Now, divide both sides by 28 to find \(S_2\):
\[S_2 = \frac{1008}{28}\]
Performing the division:
\[S_2 = 36\]
Therefore, the length of the shadow cast by the second pole is 36 meters.
Conclusion
Using the principle of similar triangles, we determined that the ratio of the height of a vertical object to the length of its shadow is constant at any given time. By setting up a proportion with the given information for the first pole and the height of the second pole, we successfully calculated the length of the shadow for the second pole.
| Pole | Height (m) | Shadow Length (m) | Ratio (Height/Shadow) |
|---|---|---|---|
| First Pole | 28 | 19.2 | \[\frac{28}{19.2} \approx 1.4583\] |
| Second Pole | 52.5 | 36 (Calculated) | \[\frac{52.5}{36} \approx 1.4583\] |
Revision Table: Pole Height and Shadow Length
Here is a quick summary of the values involved in the problem:
- Height of Pole 1: 28 m
- Shadow Length of Pole 1: 19.2 m
- Height of Pole 2: 52.5 m
- Shadow Length of Pole 2: 36 m
Additional Information: Similar Triangles in Geometry
Similar triangles are triangles that have the same shape but may be different in size. The key properties of similar triangles are:
- Corresponding angles are equal.
- Corresponding sides are in proportion (the ratio of corresponding sides is constant).
Problems involving heights and shadows of objects at the same time are classic examples where similar triangles are applied because the sun's rays hitting the tops of objects at the same time create the same angle with the horizontal ground, leading to similar right-angled triangles. This concept is useful for indirectly measuring heights or distances that are difficult to measure directly.
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