The problem involves using similar triangles to find the height of a building. Here's the given information:

• A pole is 2.9 meters tall and casts a shadow that is 1.2 meters long.
• At the same time, a nearby building casts a shadow that is 36.25 meters long.
• We need to find the height of the building.

Since both the pole and the building are casting shadows under the same lighting conditions, the ratio of the height of the object to the length of its shadow will be the same for both the pole and the building. Thus, we can set up a proportion:

$\frac{\text{Height of the pole}}{\text{Length of the pole's shadow}} = \frac{\text{Height of the building}}{\text{Length of the building's shadow}}$

Substituting the known values:

$\frac{2.9}{1.2} = \frac{\text{Height of the building}}{36.25}$

Let $H$ represent the height of the building:

$\frac{2.9}{1.2} = \frac{H}{36.25}$

Now, solve for $H$:

$H = \frac{2.9 \times 36.25}{1.2}$

$H = \frac{105.125}{1.2}$

$H \approx 87.6 \, \text{meters}$

Rounding to the nearest meter, the height of the building is approximately 88 meters.

Would you like any further details or clarification on this solution?

Here are five related questions to expand on this topic:

1. How can you apply the concept of similar triangles to real-life situations?
2. What would happen to the shadow lengths if the sun's angle changes during the day?
3. Can you calculate the angle of elevation of the sun using the shadow length and height of the pole?
4. How does the ratio of shadow to height help in indirect measurements in construction?
5. What would be the shadow length of a 5-meter tall pole under the same conditions?

Tip: To solve similar triangle problems, always remember that corresponding sides of similar triangles have the same ratio!