Let's break down the problem shown in the image:

• The rectangle $ABCD$ has dimensions of $36 \, \text{cm} \times 28 \, \text{cm}$.
• Inside the rectangle, there are 4 congruent semi-circles arranged symmetrically, each with a diameter of $10 \, \text{cm}$.
• The semi-circles are connected at the center of the rectangle, and the task is to find the area of the shaded region (i.e., the area inside the rectangle but outside the four semi-circles).

Approach:

1. Area of the rectangle: $\text{Area of rectangle} = 36 \times 28 = 1008 \, \text{cm}^2$

2. Area of the full circle formed by the semi-circles: Since the diameter of each semi-circle is $10 \, \text{cm}$, the radius $r$ of each semi-circle is: $r = \frac{10}{2} = 5 \, \text{cm}$ The full circle formed by the four semi-circles has the same radius, so the area of the full circle is: $\text{Area of circle} = \pi r^2 = \pi (5)^2 = 25\pi \, \text{cm}^2$

3. Shaded area: The shaded region is the area of the rectangle minus the area of the circle: $\text{Shaded area} = \text{Area of rectangle} - \text{Area of circle}$ $\text{Shaded area} = 1008 - 25\pi$ Using $\pi \approx 3.1416$: $\text{Shaded area} \approx 1008 - 25 \times 3.1416 \approx 1008 - 78.54 = 929.46 \, \text{cm}^2$

Thus, the area of the shaded region is approximately 929.46 cm².

Would you like further clarification, or details on any of the steps?

Related questions:

1. How would the solution change if the semi-circles had different radii?
2. Can you calculate the perimeter of the entire figure including the semi-circles?
3. What is the area if the shape were composed of quarter circles instead of semi-circles?
4. How would you calculate the shaded area if there were more semi-circles?
5. What if the rectangle had different dimensions while keeping the semi-circles' dimensions fixed?

Tip: When solving geometry problems involving multiple shapes, it's often useful to break down the problem into simple areas or perimeters and subtract or add them as necessary.