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En un mismo momento del día se miden las sombras de un edificio y un árbol que son de 14 y 2.5 metros respectivamente. Calculadora la altura del edificio si la altura del árbol es de 3.5 metros

To solve this problem, we can use the concept of similar triangles. Both the tree and the building cast shadows due to sunlight. Since the measurements were taken at the same time of day, the angles of elevation from the ends of the shadows to the tops of the objects (tree and building) are the same. Therefore, the triangles formed are similar.

### Information given:
1. Height of the tree, \( h_{\text{tree}} = 3.5 \) meters.
2. Shadow length of the tree, \( s_{\text{tree}} = 2.5 \) meters.
3. Shadow length of the building, \( s_{\text{building}} = 14 \) meters.
4. Height of the building, \( h_{\text{building}} = ? \) (to be calculated).

### Setting up the proportion:
Since the triangles are similar, the ratio of the height of the tree to the length of its shadow is the same as the ratio of the height of the building to the length of its shadow.

\[
\frac{h_{\text{tree}}}{s_{\text{tree}}} = \frac{h_{\text{building}}}{s_{\text{building}}}
\]

Substituting the known values:

\[
\frac{3.5}{2.5} = \frac{h_{\text{building}}}{14}
\]

### Solving for \( h_{\text{building}} \):

\[
h_{\text{building}} = \frac{3.5}{2.5} \times 14
\]

\[
h_{\text{building}} = 1.4 \times 14
\]

\[
h_{\text{building}} = 19.6 \text{ meters}
\]

### Conclusion:
The height of the building is 19.6 meters.

Would you like more details or have any questions?

### Related Questions:
1. How would the height of the building change if the shadow length increased to 15 meters?
2. What would be the height of the building if the tree's shadow was 3 meters long?
3. How can the concept of similar triangles be used in real-life situations?
4. What would happen if the measurements were taken at different times of the day?
5. How can we calculate the angle of elevation of the sun using these measurements?

### Tip:
Always ensure that the measurements (shadows and heights) are taken simultaneously to maintain the accuracy of calculations using similar triangles.

Learn how to calculate the height of a building using shadow lengths and the concept of similar triangles. This problem involves determining the height of a building when given the height and shadow length of a tree. Suitable for students in Grades 7-9.

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