The combined area of a square and a rectangle is 124 square centimeters. The width of the rectangle is 4 centimeters more than the length of a side of the square, and the length of the rectangle is 2 centimeters more than its width. Find the dimensions of the square and the rectangle.
s = side of square
(s^2) + [(s + 4)(s + 6)] = 124
Let's assume the side length of the square is x centimeters.
The area of a square is given by A_square = x^2 square centimeters.
Let's also assume the width of the rectangle is w centimeters.
Therefore, the length of the rectangle is (w + 2) centimeters.
Given that the width of the rectangle is 4 centimeters more than the length of a side of the square, we can write the equation as:
w = x + 4. Equation (1)
Also, the length of the rectangle is 2 centimeters more than its width, so we can write the equation as:
l = w + 2. Equation (2)
The area of the rectangle is given by A_rectangle = l * w.
We know that the combined area of the square and the rectangle is 124 square centimeters, so we can write the equation as:
A_square + A_rectangle = 124.
Substituting the values of A_square, A_rectangle, l, and w from the equations above, we get:
x^2 + (w + 2) * w = 124.
Substituting the value of w from equation (1), we get:
x^2 + (x + 4 + 2) * (x + 4) = 124.
Simplifying the equation:
x^2 + (x + 6) * (x + 4) = 124.
Expanding the equation:
x^3 + 10x^2 + 24x + 24 = 124.
Rearranging the equation:
x^3 + 10x^2 + 24x - 100 = 0.
Now, we need to solve this cubic equation to find the value of x.
We can use numerical or graphical methods to find the approximate value of x.
To find the dimensions of the square and the rectangle, we can start by assigning variables to represent the unknowns.
Let's say:
x = length of a side of the square
w = width of the rectangle
l = length of the rectangle
We're given that the width of the rectangle is 4 centimeters more than the length of a side of the square, so we can set up an equation:
w = x + 4
We're also given that the length of the rectangle is 2 centimeters more than its width, so we can set up another equation:
l = w + 2
Now, let's calculate the area of the square and the rectangle:
Area of the square = x * x = x^2
Area of the rectangle = w * l = (x + 4) * (x + 4 + 2) = (x + 4) * (x + 6)
We're told that the combined area of the square and the rectangle is 124 square centimeters:
x^2 + (x + 4) * (x + 6) = 124
Now, we can solve this quadratic equation to find the value of x. Once we find x, we can substitute it back into the equations to find the values of w and l.
Let's simplify the equation:
x^2 + (x + 4) * (x + 6) = 124
x^2 + (x^2 + 10x + 24) = 124
2x^2 + 10x + 24 - 124 = 0
2x^2 + 10x - 100 = 0
Now we can factor or use the quadratic formula to solve for x. Once we find x, we can substitute it back into the equations w = x + 4 and l = w + 2 to find the values of w and l.
I hope this explanation helps you understand the process of finding the dimensions of the square and the rectangle.