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The problem states that the lateral area \(A\) of a geometric solid is directly proportional to its apothem (\(a\)) and the perimeter of the base (\(P\)). Mathematically, this can be expressed as:

\[
A \propto a \cdot P
\]

Thus, we can introduce a constant of proportionality \(k\) and write the equation as:

\[
A = k \cdot a \cdot P
\]

### Step 1: Find \(k\) using the given data
We are told that:
- When the apothem \(a = 5 \, \text{m}\),
- The perimeter \(P = 12 \, \text{m}\),
- The lateral area \(A = 600 \, \text{m}^2\).

Substituting these values into the equation:

\[
600 = k \cdot 5 \cdot 12
\]

Solving for \(k\):

\[
600 = 60k \implies k = \frac{600}{60} = 10
\]

### Step 2: Use \(k\) to find the new area
Now, we are asked to find the area when:
- The apothem \(a = 0.5 \, \text{m}\),
- The perimeter \(P = 100 \, \text{m}\).

Using the equation \(A = k \cdot a \cdot P\) with \(k = 10\):

\[
A = 10 \cdot 0.5 \cdot 100 = 10 \cdot 50 = 500 \, \text{m}^2
\]

### Final Answer:
The area is \( \boxed{500 \, \text{m}^2} \), which corresponds to option **B**.

Would you like further clarification on any step?

### Related questions:
1. What happens if the apothem or perimeter is halved?
2. How would the area change if the apothem and perimeter doubled?
3. Can you express the lateral area in terms of just the ratio between apothem and perimeter?
4. How does the constant of proportionality affect the lateral area?
5. How would the area calculation change for a different shape (e.g., a prism or cone)?

**Tip:** Always double-check if the relation between variables involves direct or inverse proportionality.

El área lateral de un sólido geométrico es directamente proporcional a su apotema y al perímetro de la base. Si el área es 600 m², cuando el apotema es 5 m y el perímetro de la base 12 m. Halle el área cuando el apotema es 0.5 m y el perímetro de la base 100 m.

Learn how to find the lateral area of a geometric solid when the apothem and perimeter are directly proportional. Given the area is 600 m² with an apothem of 5 m and base perimeter of 12 m, calculate the area when the apothem is 0.5 m and perimeter is 100 m.

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