At 3:00 PM a man 135 cm tall casts a shadow 142 cm long. At the same time, a tall building nearby casts a shadow 165 m long. How tall is the building?


The building is meters tall.
Round to two decimal places if necessary.

To find the height of the building, we can use the concept of similar triangles. The man's height and his shadow, as well as the building's height and its shadow, form two similar triangles. Since the triangles are similar, the ratio of the man's height to his shadow will be the same as the ratio of the building's height to its shadow.

### Step 1: Write down the ratios
Let \( h \) be the height of the building in meters. The ratios are:
\[
\frac{\text{Man's height}}{\text{Man's shadow}} = \frac{\text{Building's height}}{\text{Building's shadow}}
\]
Substituting the given values:
\[
\frac{135 \, \text{cm}}{142 \, \text{cm}} = \frac{h \, \text{meters}}{165 \, \text{meters}}
\]

### Step 2: Solve for \( h \)
\[
h = \frac{135}{142} \times 165
\]
Let's calculate this:

\[
h = \frac{135 \times 165}{142} = \frac{22275}{142} \approx 156.87 \, \text{meters}
\]

### Final Answer:
The building is approximately \( 156.87 \) meters tall.

Would you like further details on any step or have any questions?

Here are 5 related questions:
1. How can you apply the concept of similar triangles in real-life situations?
2. What other methods can be used to find the height of a building without direct measurement?
3. How do you convert between different units when solving similar triangle problems?
4. What is the importance of rounding in real-world applications?
5. How can trigonometric ratios be used in similar problems?

**Tip:** When dealing with similar triangles, always ensure the units are consistent before setting up the ratio.

Learn how to calculate the height of a building using the concept of similar triangles. Given the height of a man and his shadow, and the shadow of a nearby building, find the building's height using proportional reasoning. Step-by-step solution provided.

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