This problem involves determining the area of a specific trapezoidal region in an isosceles triangle. Let me break it down step by step:

Information from the problem:

1. Isosceles triangle dimensions:

   • Side length ($s$) = $24 \, \text{cm}$,
   • Total area of the triangle ($A$) = $220 \, \text{cm}^2$.

2. Division of the triangle:

   • The triangle is divided into four sections by lines parallel to the base, such that the height of the triangle is divided into four equal parts.

3. Goal:

   • Find the area of the shaded section.

Step-by-step solution:

Step 1: Calculate the height of the triangle ($h$):

The formula for the area of a triangle is: $A = \frac{1}{2} \cdot \text{base} \cdot \text{height}.$ Let the base of the triangle be $b$. Since the triangle is isosceles, the height ($h$) can be calculated from: $h = \frac{2A}{b}.$

From the problem, the total area ($A$) is $220 \, \text{cm}^2$. We need to calculate $b$ first.

Step 2: Express dimensions of the sections using height proportions:

Each horizontal division creates similar smaller triangles, with areas proportional to the squares of their heights. By dividing the height into four equal parts, the heights of the triangles from bottom to top will have proportions: [ \left(\frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4} \right).