A vertical pole of length `7.5 m ` casts a shadow 5 long on the ground and at the same time a tower casts a shadow 24 m long. Find the height of the tower.

Answer

To find the height of the tower based on the given information, we can use the concept of similar triangles. Here's the step-by-step solution:

### Step 1: Understand the relationship
We have a vertical pole and a tower casting shadows. The height of the pole is 7.5 m and its shadow is 5 m long. The tower casts a shadow of 24 m. Since both the pole and the tower are vertical and the sun's rays are the same for both, the triangles formed by the pole and its shadow, and the tower and its shadow are similar.

### Step 2: Set up the proportion
Using the property of similar triangles, we can set up the following proportion:
\[ \frac{\text{Height of the pole}}{\text{Length of the shadow of the pole}} = \frac{\text{Height of the tower}}{\text{Length of the shadow of the tower}} \]
Let the height of the tower be \( h \). Thus, we can write:
\[ \frac{7.5}{5} = \frac{h}{24} \]

### Step 3: Cross-multiply to solve for \( h \)
Cross-multiplying gives us:
\[ 7.5 \times 24 = 5 \times h \]
Calculating \( 7.5 \times 24 \):
\[ 7.5 \times 24 = 180 \]
So, we have:
\[ 180 = 5h \]

### Step 4: Solve for \( h \)
Now, divide both sides by 5 to find \( h \):
\[ h = \frac{180}{5} = 36 \]

### Conclusion
The height of the tower is \( 36 \) meters.

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