A rectangle $PQRS$ has dimensions $20$ cm by $(10+10)$ cm $= 20$ cm. A square of side $x$ cm has been cut from the rectangle. The area of the shaded portion is $484$ cm$^2$. 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 has been cut from the rectangle. The area of the shaded portion is $484$ cm$^2$. Find the value of $x$.

2. Solution Steps

The area of rectangle $PQRS$ is given by

$$\text{Area}_{PQRS} = \text{length} \times \text{width} = 20 \times (10+10) = 20 \times 20 = 400 \text{ cm}^2$$

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

The area of the shaded portion is the area of the rectangle minus the area of the square. Thus,

$$\text{Area}_{\text{shaded}} = \text{Area}_{PQRS} - \text{Area}_{\text{square}}$$

We are given that the area of the shaded region is $484 \text{ cm}^2$. Therefore,

$$484 = 400 - x^2$$

Rearranging the equation, we get

$$x^2 = 400 - 484$$

However, the value of $x^2$ cannot be negative since $x$ represents a length. Thus there must be an error.

The length of the rectangle $PQRS$ is $20$ cm.

The width of the rectangle is $10+10 = 20$ cm.

The area of the rectangle is $20 \times 20 = 400$ cm$^2$.

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

The area of the shaded portion is given as $484$ cm$^2$.

The correct equation should be:

Area of shaded region = Area of rectangle - Area of squares (There are two squares)

Then, Area of two squares $= 2x^2$. The shaded region's area is Area of rectangle $PQRS$ $-$ Area of the small squares.

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

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

Area of shaded portion $= 400 - 2x^2 = 484$ (ERROR). The shaded region's area is less than the total area of rectangle.

It is written that the diagram shows a rectangle $PQRS$ from which a square of side $x$ cm has been cut. But there are two squares. So we must subtract $x^2 + x^2 = 2x^2$. Thus,

$400 - x^2 = 484$, so $-x^2 = 84$, which is impossible. The figure must have more squares.

Another interpretation:

Area of rectangle = $20 \times 20 = 400$

Area of one square = $x^2$

The area of the shaded portion = $400 - x^2 = 484$ means $x^2 = 400 - 484 = -84$. Something is wrong.

The shaded region's area must be less than the rectangle's area,

4

0

0. There appears to be two squares of $x$ cm each.

Let $PQ = SR = 20$. Let $PS=QR = 20$.

The area of rectangle $PQRS = 20 \times 20 = 400$.

The area of the squares (total) = Area Square 1 + Area Square 2 = $x^2 + x^2 = 2x^2$.

Area of shaded region = $400 - 2x^2$.

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

$400 - x^2 = 484$, so $x^2 = 400 - 484 = -84$, which is impossible since $x$ must be positive.

Let's assume there's a typo in the question.

$400 - 2x^2 = 484$. Then $-2x^2 = 84$, $2x^2 = -84$, also impossible.

The area of rectangle $PQRS = 20*(10+10) = 400$. Area of square $= x^2$.

Then $400-x^2 = 484$.

Consider that the area of the shaded portion is actually $316$ rather than $484$. Then,

$400-x^2 = 316$

$x^2 = 400 - 316 = 84$.

$x = \sqrt{84} = 2\sqrt{21} \approx 9.165$.

$400 - 2x^2 = 484$. Then $400 - 2x^2 = \text{area of the shaded region}$. The maximum possible area $= 400$.

Then if shaded region area = 316, then

$400 - 2x^2 = 316$, $2x^2 = 400-316 = 84$, so $x^2 = 42$. $x = \sqrt{42} = 6.48$ cm.

Let's re-examine the problem: A rectangle $PQRS$ from which a SQUARE of SIDE $x$ cm has been cut.

If the shaded portion is $484 \text{ cm}^2$, then $400 - x^2 = 484$. This gives $x^2 < 0$.

So the figure must have multiple cutouts. It appears there are two squares cut out. Area = rectangle area - 2 squares.

So $400 - 2x^2 = 484$. Impossible.

Let the area be something like

2

1

6. $400 - 2x^2 = 216$. Then $2x^2 = 184$, so $x^2 = 92$. $x=\sqrt{92}$.

If the shaded region's area = 316, then $400 - 2 x^2 = 316$.

$2 x^2 = 84$. $x^2 = 42$.

$x = \sqrt{42} \approx 6.48$.

3. Final Answer

There appears to be an error in the problem. The area of the shaded region must be less than 400 cm$^2$, which is the area of the rectangle $PQRS$. If the problem intended for us to consider there are *two* squares of side $x$ removed, then the area of the two squares is $2x^2$. Let's assume the area of the shaded region is intended to be $316 \text{ cm}^2$, Then $400 - 2x^2 = 316$. Solving for $x$, $2x^2 = 400 - 316 = 84$, thus $x^2 = 42$, and $x = \sqrt{42} \approx 6.48$ cm. But we have the area of the shaded region given as

4

8

4. In this case, there is no solution.

Final Answer: There is no real solution for x. Assuming the area of the shaded region is 316 cm$^2$ and there are two squares, the value of x would be $\sqrt{42} \approx 6.48$.