A. The perimeter of a rectangle is given by the formula:
Perimeter = 2(length + width)
In this case, the length of the rectangle is (x+4) cm, and the width is x cm.
Therefore, the expression for the perimeter of the rectangle would be:
Perimeter = 2[(x+4) + x]
= 2[2x + 4]
= 4x + 8 cm
B. The length of a side of a square with the same perimeter can be calculated by dividing the perimeter by 4 (since a square has four equal sides).
In this case, the perimeter is 4x + 8 cm, so the expression for the length of the side of the square would be:
Side length of square = (4x + 8)/4
= x + 2 cm
To find the value of x, we can use the information about the sum of the areas of the square and the rectangle being 94 cm².
The area of a rectangle is given by multiplying the length and width:
Area of rectangle = length * width
= (x+4) * x
= x² + 4x cm²
Since the area of the square is also 94 cm², we can set up the equation:
Area of rectangle + Area of square = 94
(x² + 4x) + (x + 2)² = 94
Simplifying this equation will help us find the value of x.
Expanding the squared term:
(x² + 4x) + (x² + 4x + 4) = 94
Combining like terms:
2x² + 8x + 4 = 94
Subtracting 94 from both sides:
2x² + 8x + 4 - 94 = 0
2x² + 8x - 90 = 0
Now, we can factor this equation or use the quadratic formula to solve for x.
Factoring method:
Divide every term by 2 to simplify the equation:
x² + 4x - 45 = 0
(x + 9)(x - 5) = 0
Setting each factor equal to zero:
x + 9 = 0 or x - 5 = 0
Solving for x in each equation:
x = -9 or x = 5
However, since the length and width of the rectangle cannot be negative, we discard x = -9 as an extraneous solution.
Therefore, the value of x is 5.