A RECTANGLE HAS A WIDTH OF X CENTIMETERS AND A PERIMETER OF 8X CENTIMETERS. A SQUARE HAS SIDES OF LENGTH 1/4 THAT OF THE RECTANGLE

A) FIND THE LENGTH OF THE RECTANGLE
B) FIND THE PERIMETER OF THE SQUARE
C) FIND HOW MANY CCENTIMETERS GREATER THE RECTANGLE IS THAN THE SQUARES PERIMETER IF X=4
D) FIND HOW MANY SQUARE CENTIMETERS GREATER THEN RECTANGLE'S AREA IS THAN THE SQUARES AREA IF X=4

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A) To find the length of the rectangle, we need to use the given information that the rectangle has a width of x centimeters and a perimeter of 8x centimeters.

The formula to calculate the perimeter of a rectangle is:
Perimeter = 2 * (Length + Width)

Since we know that the perimeter is 8x and the width is x, we can substitute these values into the formula and solve for the length:
8x = 2 * (Length + x)
Divide both sides by 2:
4x = Length + x
Subtract x from both sides:
4x - x = Length
Simplifying the equation:
3x = Length

Therefore, the length of the rectangle is 3x centimeters.

B) Now, let's find the perimeter of the square. The square's sides have a length of 1/4 that of the rectangle. Since the rectangle's width is x, the square's sides would be 1/4 * x:

Side of the square = (1/4) * x

The perimeter of a square is calculated by multiplying the length of one side by 4:
Perimeter = 4 * (Side of the square) = 4 * (1/4 * x) = x

So, the perimeter of the square is x centimeters.

C) To find how many centimeters greater the rectangle is than the square's perimeter when x = 4, we need to calculate the difference between the length of the rectangle and the perimeter of the square.

Rectangle's length = 3x = 3 * 4 = 12 centimeters
Square's perimeter = x = 4 centimeters

The difference is:
Rectangle's length - Square's perimeter = 12 - 4 = 8 centimeters

Therefore, when x = 4, the rectangle is 8 centimeters greater than the square's perimeter.

D) To find how many square centimeters greater the rectangle's area is than the square's area when x = 4, we need to calculate the difference between their areas.

Area of a rectangle is calculated by multiplying its length and width:
Area of rectangle = Length * Width = (3x) * x = 3x^2

Area of a square is calculated by squaring the length of its sides:
Area of square = (Side of the square)^2 = (x)^2 = x^2

The difference in their areas is:
Area of rectangle - Area of square = 3x^2 - x^2 = 2x^2

When x = 4:
Difference in areas = 2 * (4)^2 = 2 * 16 = 32 square centimeters

Therefore, when x = 4, the rectangle's area is 32 square centimeters greater than the square's area.