The area of a rectangle is 54 centimeters squared. The side lengths are 2x+1 and x+2. What is the measure of each side
2(2x + 1) + 2(x + 2) = 54
4x + 2 + 2x + 4 = 54
6x + 6 = 54
6x = 48
x = 8
Take it from there.
To find the length of each side of the rectangle, we need to set up an equation using the given information.
The area of a rectangle is found by multiplying the length by the width. We are given that the area of the rectangle is 54 cm^2, so we can set up the following equation:
(2x + 1)(x + 2) = 54
Now we can solve this equation to find the value of x, which will give us the measurements of the sides of the rectangle.
Expanding the equation, we get:
2x^2 + 4x + x + 2 = 54
Combining like terms, we get:
2x^2 + 5x + 2 = 54
Moving the constant term to the other side, we get:
2x^2 + 5x - 52 = 0
Now, we have a quadratic equation in standard form. We can solve this equation by factoring, completing the square, or using the quadratic formula.
Since the quadratic equation doesn't factor easily, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
In this equation, a = 2, b = 5, and c = -52. Substituting these values into the quadratic formula, we get:
x = (-5 ± √(5^2 - 4*2*(-52)))/(2*2)
Simplifying inside the square root, we have:
x = (-5 ± √(25 + 416))/(4)
x = (-5 ± √(441))/(4)
x = (-5 ± 21)/(4)
Now, we have two solutions for x:
1. x = (-5 + 21)/(4) = 16/4 = 4
2. x = (-5 - 21)/(4) = -26/4 = -13/2
Since the length of a side cannot be negative, we will consider the positive value of x, which is x = 4.
Now, we can substitute this value of x back into the expressions for the side lengths of the rectangle:
Length = 2x + 1 = 2(4) + 1 = 8 + 1 = 9 cm
Width = x + 2 = 4 + 2 = 6 cm
Therefore, the measure of each side of the rectangle is 9 cm and 6 cm.