A rectangle $PQRS$ with length $20$ cm and width $10+10 = 20$ cm has a square of side $x$ cm cut out from it. The area of the shaded portion is $484$ cm$^2$. We need to find the value of $x$.

1. Problem Description

A rectangle $PQRS$ with length $20$ cm and width $10+10 = 20$ cm has a square of side $x$ cm cut out from it. The area of the shaded portion is $484$ cm$^2$. We need to find the value of $x$.

2. Solution Steps

The area of the rectangle $PQRS$ is given by:

$A_{rectangle} = length \times width = 20 \times (10 + 10) = 20 \times 20 = 400$ cm$^2$

The area of the square that is cut out is given by:

$A_{square} = x^2$

The area of the shaded portion is the area of the rectangle minus the area of the square:

$A_{shaded} = A_{rectangle} - A_{square}$

We are given that $A_{shaded} = 484$ cm$^2$. Therefore,

$484 = 400 - x^2$

$x^2 = 400 - 484$

This is incorrect. The correct setup should be:

The area of the rectangle $PQRS$ is $20 \times 20 = 400$ cm$^2$. The area of the square cut out is $x^2$ cm$^2$.

The area of the shaded portion is $400 - x^2 = 484$ cm$^2$. However, it looks like two equal squares, each having an area of x^2 were cut out, so the equation is

$400 - x^2=484$

However, this is wrong, as 400 - x^2 < 400, so x^2 must be larger than

0.

The full length of SR is 10cm + 10cm = 20cm. The length of PS is 20cm.

Therefore the area of the whole rectangle is 20cm * 20cm = 400cm^2

From this we remove one area, x^

2. Therefore $400 - x^2$ must equal $484cm^2$.

However this is clearly wrong, as $400 - x^2 < 400$. This means we must have the wrong dimensions.

Rectangle PQRS is Length = 20cm. Width = 10+10 = 20cm. So the area is $20cm * 20cm=400cm^2$

Area of square that is removed is $x^2$. Area that is left is 400- $x^2$. The question statement is that $484 = 400-x^2$

The picture must be drawn in a different way.

Let's assume there's one such square cut off from the rectangle.

Then $Area_{shaded}=400-x^2 = 484$. This doesn't make sense

Rectangle Area :20 x 20 = 400cm^2

Square Area: x^2

Shaded area= 484 cm^

2. This problem is incorrect

Let's examine the figure carefully.

The width of the rectangle is (10+10)cm =20cm. Length = 20cm

The area of the whole rectangle = 20cm * 20cm= 400cm^

2. $Area(shaded)=484cm^2$

$A(rect) - x^2=484cm^2$

$Area(rect)=400 - x^2cm^2$.

If shaded area is 484cm^2

Then$400 - x^2=484$. This implies $x^2=400-484= -84$.

$x=\sqrt{-84}$=Impossible.

This problem has no possible value of x.

It looks like there are two cutouts: (10*x) on the right side and (10 *x) on the left side.

$A_rect = 20 * 20 = 400$

Then $Area_rect - Area_cutouts = 484$

3. Final Answer

There is no solution for x. The given area of the shaded region is incorrect.

The correct equation setup for one square cutout:

$Area(shaded) = 400 - x^2$

$484=400-x^2 => x^2=400-484 = -84 cm^2$

$x= \sqrt{-84}$. There are no real solutions.