The length and width of a rectangle are (x+4) cm and x cm respectively. Write down expressions for:

A. The perimeter of the rectangle
B. The length of the side of a square with the same perimeter

If the sum of the areas of the square and the rectangle is 94 cm^2 , find x. Explain your working and solution

a) P = 2(x+4) + 2x = 4x + 8

b) if this had been a square, then each side would have been (4x+8)/4 = x+2

x(x+4) + (x+2)^2 = 94
x^2 + 4x + x^2 + 4x + 4 = 94
2x^2 + 8x - 90 = 0
x^2 + 4x - 45 = 0
(x+9)(x-5) = 0
x = a negative or x = 5

check:
rectange was 5 by 9 for an area of 45
square was 7 by 7 for an area of 49
total area is 45+49 = 94 ,
all is good!

To solve this problem, we'll first write down the expressions for the perimeter and the side length of the square. Then, we'll set up an equation using the given information about the sum of the areas to find the value of x.

A. The perimeter of a rectangle is given by the sum of all its sides. In this case, the rectangle has a length of (x+4) cm and a width of x cm. So, the perimeter can be calculated as follows:

Perimeter = 2(length + width)
= 2((x+4) + x)
= 2(2x+4)
= 4x + 8 cm

B. The length of the side of a square with the same perimeter is equal to the perimeter divided by 4. Therefore, the expression for the length of the side of a square with the same perimeter is:

Side length of square = Perimeter / 4
= (4x + 8) / 4
= x + 2 cm

Now, let's proceed to set up an equation using the given information about the sum of areas:

Area of rectangle + Area of square = 94 cm^2

The area of a rectangle is given by length multiplied by width, which is (x+4) * x.
The area of a square is given by the side length squared, which is (x + 2)^2.

So, we can write the equation as follows:

(x+4) * x + (x+2)^2 = 94

Expanding the equation yields:

x^2 + 4x + x^2 + 4x + 4 = 94

Combining like terms gives:

2x^2 + 8x + 4 = 94

Simplifying the equation:

2x^2 + 8x - 90 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:

2(x^2 + 4x - 45) = 0

(x^2 + 4x - 45) = 0

Factoring the quadratic expression:

(x + 9)(x - 5) = 0

Setting each factor equal to zero and solving for x:

x + 9 = 0 or x - 5 = 0

x = -9 or x = 5

Since lengths cannot be negative, the solution is x = 5 cm.

Therefore, the length and width of the rectangle are 9 cm and 5 cm, respectively.