The side of a square equals the length of a rectangle. The width of the rectangle is 4 meters longer than its length. The sum of the areas of the square and the rectangle is 48 square centimeters. Find the side of the square
square --- x by x
rectangle -- x by x+4
x^2 + x(x+4) = 48
2x^2 + 4x - 48 = 0
x^2 + 2x - 24 = 0
(x+6)(x-4) = 0
x = -6 or x = 4, but a side cannot be negative
so the square is 4 by 4
each unit square of the rectangle below represents 1 square inch name the rectangle that show perimeter and name the permeter
Let's start by assigning variables to the given information. Let:
- "x" represent the length of the rectangle in meters,
- "x + 4" represent the width of the rectangle in meters, and
- "s" represent the side length of the square in centimeters.
We know that the side length of the square equals the length of the rectangle, so we can set up the equation:
s = x
The area of a rectangle is given by the formula: length × width. Therefore, the area of the rectangle is:
Rectangle area = x × (x + 4)
The area of a square is given by the formula: side length × side length. Therefore, the area of the square is:
Square area = s × s = x × x
The problem states that the sum of the areas of the square and rectangle is 48 square centimeters. We can express this as an equation:
Rectangle area + Square area = 48
Substituting the values from above into the equation, we get:
x × (x + 4) + x × x = 48
Simplifying the equation:
x^2 + 4x + x^2 = 48
2x^2 + 4x = 48
2x^2 + 4x - 48 = 0
Now, we can solve this quadratic equation to find the value of "x". We can either factor, complete the square, or use the quadratic formula to solve it. Let's use factoring in this case.
First, we factor out a common factor of 2 from the equation:
2(x^2 + 2x - 24) = 0
Then, we can factor the quadratic expression inside the parentheses:
2(x - 4)(x + 6) = 0
Setting each factor equal to zero:
x - 4 = 0 or x + 6 = 0
Solving for "x":
x = 4 or x = -6
Since we're dealing with lengths, the value of "x" cannot be negative. Therefore, we take x = 4 as the valid solution.
Finally, we substitute the value of "x" back into the equation for the side length of the square:
s = x = 4
Therefore, the side length of the square is 4 meters.
To find the side of the square, we need to set up an equation based on the given information.
Let's assume that the side of the square is "x" meters.
According to the given information, the length of the rectangle is also equal to "x" meters.
The width of the rectangle is 4 meters longer than its length, so the width would be "x + 4" meters.
The area of a square is calculated by multiplying the side length by itself. Therefore, the area of the square would be "x * x" or "x^2" square meters.
The area of a rectangle is calculated by multiplying its length by its width. Therefore, the area of the rectangle would be "x * (x + 4)" square meters.
According to the problem, the sum of the areas of the square and the rectangle is 48.
So, we can set up the equation:
x^2 + x(x + 4) = 48
Simplifying the equation:
x^2 + x^2 + 4x = 48
Combining like terms:
2x^2 + 4x - 48 = 0
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring the quadratic equation:
2x^2 + 4x - 48 = 0
2(x^2 + 2x - 24) = 0
2(x + 6)(x - 4) = 0
Setting each factor equal to zero:
x + 6 = 0 or x - 4 = 0
x = -6 or x = 4
Since the length cannot be negative, we can ignore the solution x = -6.
Therefore, the side of the square, x, is equal to 4 meters.