The length and width of a rectangle are (x+4)cm and xcm 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 94cm(square), find x
Pls include workings thx
p = 2(width+length) = 2(x+4+x) = 2(2x+4) = 4x+8
s = p/4 = (4x+8)/4 = x + 2
94 = x(x+4) + (x + 2)^2
94 = x^2 + 4x + x^2 + 4x + 4
94 = 2x^2 + 8x + 4
2x^2 + 8x - 90 = 0
x^2 + 4x - 45 = 0
(x+9)(x-5) = 0
x = 5
rectangle is 5x9; p=28 a=45
square = 7x7; p=28 a=49
45+49 = 94
A. The perimeter of the rectangle is the sum of all sides. Since the length is (x + 4) cm and the width is x cm, the perimeter (P) can be expressed as:
P = 2(length + width)
P = 2((x + 4) + x)
P = 2(2x + 4)
P = 4x + 8
B. The perimeter of a square is defined as 4 times the length of one side (s). Since we already have an expression for the perimeter of the rectangle (4x + 8), we can set it equal to the perimeter of the square and solve for the length of one side (s):
4s = 4x + 8
s = (4x + 8)/4
s = x + 2
The length of the side of a square with the same perimeter is x + 2 cm.
Now, let's find the value of x using the information given about the sum of the areas of the square and the rectangle:
The area of the rectangle (A_rect) is given by the length multiplied by the width:
A_rect = length * width
A_rect = (x + 4) * x
A_rect = x^2 + 4x
Since the area of the square (A_sq) is the length of one side squared, and the length of one side is x + 2 cm, the area of the square can be expressed as:
A_sq = (x + 2)^2
A_sq = x^2 + 4x + 4
Given that the sum of the areas is 94 cm^2, we can set up an equation:
A_rect + A_sq = 94
x^2 + 4x + x^2 + 4x + 4 = 94
2x^2 + 8x + 4 = 94
2x^2 + 8x - 90 = 0
Divide the entire equation by 2 to simplify:
x^2 + 4x - 45 = 0
We can now solve for x by factoring or using the quadratic formula. Factoring, we have:
(x + 9)(x - 5) = 0
Setting each factor equal to zero:
x + 9 = 0 --> x = -9 (Discard since we're dealing with lengths)
or
x - 5 = 0 --> x = 5
Therefore, x = 5 cm.
To summarize:
A. The perimeter of the rectangle is 4x + 8 cm.
B. The length of the side of a square with the same perimeter is x + 2 cm.
The value of x is 5 cm.
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.