A rectangle $PQRS$ has dimensions $20$ cm by $(10+10)$ cm $= 20$ cm. A square of side $x$ cm is cut out from the rectangle. The area of the shaded portion is given as $484$ cm$^2$. We need to find the value of $x$.

1. Problem Description

A rectangle $PQRS$ has dimensions $20$ cm by $(10+10)$ cm $= 20$ cm. A square of side $x$ cm is cut out from the rectangle. The area of the shaded portion is given as $484$ 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 $PQRS$ = length $\times$ width = $20 \times (10+10) = 20 \times 20 = 400$ cm$^2$.

The area of the square that is cut out is $x^2$ cm$^2$.

Area of shaded portion = Area of rectangle - Area of square.

We are given that the area of the shaded portion is $484$ cm$^2$.

So, $400 - x^2 = 484$ is incorrect.

Looking at the figure, we see that the length of rectangle is $10 + 10 = 20$ cm and width is $20$ cm. Also, the square that is cut off has sides equal to $x$.

Area of the rectangle = $20 \times 20 = 400$ cm$^2$.

Area of the square = $x \times x = x^2$ cm$^2$.

Area of shaded region = Area of rectangle - Area of square

$484$ = $400 + area$ which is not right, the area of shaded portion cannot be greater than the total area.

Let the rectangle be $20$ by $20$, and the square has sides $x$. The shaded area is

4

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4. $20 \times 20 - x^2 = 400-x^2$, which equals the shaded area. But 484 is greater than

4

0

0. There must be some mistake somewhere.

The length of rectangle $PQRS$ is $10 \text{ cm} + 10 \text{ cm} = 20 \text{ cm}$ and the width is $20 \text{ cm}$. So the area of the rectangle $PQRS$ is $20 \text{ cm} \times 20 \text{ cm} = 400 \text{ cm}^2$.

The area of the square cut out is $x^2 \text{ cm}^2$.

So, Area of shaded portion = Area of rectangle - Area of square

$484 = 400 - x^2$ which is wrong because we end up with $x^2 = -84$, and we want the real solution.

Upon looking at the figure, it appears the square is cut into the original rectangle. That implies

Area of rectangle - area of square = Shaded area.

$20 \times (10 + 10) - x^2 = 484$

$400 - x^2 = 484$

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

Which is not a real value.

I realized that my mistake is the value

4

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4. Let us assume instead the shaded area is $A = 384 cm^2$.

$400 - x^2 = 384$

$x^2 = 400 - 384$

$x^2 = 16$

$x = 4$ cm.

Assuming the printed value is 384 instead of 484:

Area of shaded portion = Area of rectangle $PQRS$ - Area of square

$384 = (20 \times 20) - x^2$

$384 = 400 - x^2$

$x^2 = 400 - 384$

$x^2 = 16$

$x = \sqrt{16}$

$x = 4$ cm

However, the problem states that the shaded area is $484 \text{ cm}^2$. This is not possible. If we assume the problem is printed wrong, and the shaded region is actually 384, the answer would be

4.

3. Final Answer

Due to inconsistency of information, there is no real solution. If the area was 384 cm$^2$, then $x = 4$ cm. With the information as provided, there is no answer. There appears to be a printing error on the value.