If a 7-storey building has a 28 m long shadow, the number of storeys of the building whose shadow is 48 m long is:
12
Understanding the Relationship Between Building Storeys and Shadow Length
This problem involves the relationship between the height of a building, represented by the number of storeys, and the length of the shadow it casts. At any given time of day, the angle of the sun is constant. This means that for objects standing upright, the ratio of their height to the length of their shadow is constant. We can use this principle of proportionality to solve the problem.
Think of the building and its shadow as two sides of a right-angled triangle, with the sun's rays forming the hypotenuse. For different buildings at the same time, these triangles are similar, meaning their corresponding sides are in proportion.
Setting Up the Proportion
Let:
- \(S_1\) be the shadow length of the first building (28 m)
- \(H_1\) be the 'height' of the first building, proportional to its storeys (7 storeys)
- \(S_2\) be the shadow length of the second building (48 m)
- \(H_2\) be the 'height' of the second building, proportional to its storeys (let this be \(x\) storeys)
Since the ratio of height to shadow length is constant, we can write the proportion:
\[ \frac{H_1}{S_1} = \frac{H_2}{S_2} \]
Substituting the given values:
\[ \frac{7 \text{ storeys}}{28 \text{ m}} = \frac{x \text{ storeys}}{48 \text{ m}} \]
Solving for the Unknown Number of Storeys
We need to find the value of \(x\). We can solve this proportion:
\[ \frac{7}{28} = \frac{x}{48} \]
Simplify the fraction on the left side:
\[ \frac{1}{4} = \frac{x}{48} \]
Now, to isolate \(x\), multiply both sides of the equation by 48:
\[ x = \frac{1}{4} \times 48 \]
Perform the multiplication:
\[ x = \frac{48}{4} \]
\[ x = 12 \]
So, the number of storeys of the building whose shadow is 48 m long is 12.
Conclusion on Building Storeys and Shadow Length
Based on the principle of proportionality, a building with a 48 m long shadow, under the same conditions where a 7-storey building casts a 28 m shadow, would have 12 storeys.
Revision Table: Building Shadow Calculation
| Building | Number of Storeys | Shadow Length (m) |
|---|---|---|
| First Building | 7 | 28 |
| Second Building | \(x\) | 48 |
Additional Information: Proportionality and Similar Triangles
The concept used here is based on similar triangles in geometry. When the sun shines on two vertical objects at the same time, the triangles formed by each object, its shadow, and the sun's rays are similar. Similar triangles have corresponding angles that are equal and corresponding sides that are proportional. This means the ratio of the height of an object to the length of its shadow is the same for all vertical objects in the vicinity at that moment.
This proportionality allows us to find an unknown height or shadow length if we know the corresponding measurement and the ratio from another object under the same conditions. It's a practical application of geometry.
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