The length of a rectangle is 3 m more than double the width, and the area of the rectangle is 54 m^2. Find the dimensions of the rectangle.

L = 2 W + 3

A = L * W = ( 2 W + 3 ) * W = 2 W ^ 2 + 3 W

A = 2 W ^ 2 + 3 W

54 = 2 W ^ 2 + 3 W Subtract 54 to both sides

54 - 54 = 2 W ^ 2 + 3 W - 54

0 = 2 W ^ 2 + 3 W - 54

2 W ^ 2 + 3 W - 54 = 0

The solutions are :

W = - 6

and

W = 9 / 2

Width can't be negative so :

W = 9 / 2 = 4.5 m

L = 2 W + 3 = 2 * ( 9 / 2 ) + 3 = 9 + 3 = 12 m

A = L * W = 12 * 9 / 2 = 108 / 2 = 54 m ^ 2

To find the dimensions of the rectangle, we can set up an equation using the given information.

Let's assume the width of the rectangle is represented by 'w'. Since the length is 3 meters more than double the width, we can express it as '2w + 3'.

The area of a rectangle is calculated by multiplying its length and width. Given that the area is 54 square meters, we can write the equation as:

w * (2w + 3) = 54

Now we can solve this equation to find the value of 'w', which represents the width of the rectangle.

Expanding the equation, we have:

2w^2 + 3w = 54

Rearranging the equation in standard quadratic form, we have:

2w^2 + 3w - 54 = 0

To solve this quadratic equation, we can factorize it or use the quadratic formula. Let's use the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 2, b = 3, and c = -54. Substituting these values into the quadratic formula, we get:

w = (-3 ± √(3^2 - 4 * 2 * -54)) / (2 * 2)

Simplifying further, we have:

w = (-3 ± √(9 + 432)) / 4

w = (-3 ± √(441)) / 4

w = (-3 ± 21) / 4

Solving for the positive and negative cases of the square root:

Case 1: w = (-3 + 21) / 4 = 18 / 4 = 4.5
Case 2: w = (-3 - 21) / 4 = -24 / 4 = -6

Since the width cannot be negative in this context, we discard the negative solution. Therefore, the width of the rectangle is 4.5 meters.

Now, to find the length, we substitute the value of 'w' we just found into our expression '2w + 3':

Length = 2(4.5) + 3 = 9 + 3 = 12 meters

Hence, the dimensions of the rectangle are 4.5 meters by 12 meters.