The problem describes a rectangle PQRS with dimensions 20 cm and (10+10) cm = 20 cm. Two squares of side $x$ cm are cut from the rectangle. The area of the shaded region is 484 $cm^2$. We need to find the value of $x$.

1. Problem Description

The problem describes a rectangle PQRS with dimensions 20 cm and (10+10) cm = 20 cm. Two squares of side $x$ cm are cut from the rectangle. The area of the shaded region is 484 $cm^2$. We need to find the value of $x$.

2. Solution Steps

First, we calculate the area of the rectangle PQRS.

Area of rectangle = length * width

Area of rectangle = $20 * (10+10) = 20 * 20 = 400 cm^2$

Next, we calculate the total area of the two squares that are cut out.

Area of one square = side * side = $x * x = x^2$

Area of two squares = $2 * x^2 = 2x^2$

The area of the shaded portion is the area of the rectangle minus the area of the two squares. We are given that the area of the shaded portion is 484 $cm^2$. Therefore:

Area of rectangle - Area of two squares = Area of shaded portion

$400 - 2x^2 = 484$

Now we solve for $x$.

$-2x^2 = 484 - 400$

$-2x^2 = 84$

$x^2 = -42$

Since $x^2$ cannot be negative in this real-world context (side length of a square), there might be an error in the original problem statement, probably in the area of the shaded area.

Let's assume that the area of the shaded region is something less than the total area. For example, let's assume the area of the shaded region is 316 $cm^2$.

Then $400 - 2x^2 = 316$

$-2x^2 = 316 - 400$

$-2x^2 = -84$

$x^2 = 42$

$x = \sqrt{42}$

However, returning to the original problem, if the area of the shaded region is indeed 484, the equation becomes $400 - 2x^2 = 484$, which gives $x^2 = -42$. Since we cannot have a negative value for the square of a length, there is no real solution for $x$. Let us re-examine the problem statement to see if we missed something. The total area of the rectangle is

4

0

0. The shaded area is

4

8

4. This means that the area of the cutouts is negative. This is impossible.

3. Final Answer

There is no real solution for $x$. The problem statement is incorrect.

Assuming the area of shaded region is 316 $cm^2$, then $x = \sqrt{42}$ cm.