The width of a rectangle is 3 cm less than the length. The area of the rectangle is 54 square cm. Find the width and length of the rectangle.
I'll work with the width, since the letter l and the digit 1 look similar.
if the length is 3 more than the width,
w(w+3) = 54
w^2 + 3w = 54
w^2 + 3w - 54 = 0
(w+9)(w-6) = 0
w=6 (can't have negative width)
the rectangle is 6x9
To find the width and length of the rectangle, we can set up a system of equations based on the given information.
Let's assume the length of the rectangle is "x" cm.
According to the given information, the width of the rectangle is 3 cm less than the length. So, the width would be (x - 3) cm.
The area of a rectangle is calculated by multiplying its length and width. We are given that the area is 54 square cm, so we have the equation:
Length * Width = Area
x * (x - 3) = 54
Now, let's solve this equation to find the values of x and (x - 3).
Expanding the equation, we get:
x^2 - 3x = 54
Moving all the terms to one side, we have:
x^2 - 3x - 54 = 0
Now, we can either factorize or use the quadratic formula to solve this quadratic equation. Let's use the quadratic formula:
The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -3, and c = -54.
Plugging in these values into the quadratic formula, we get:
x = (-(-3) ± sqrt((-3)^2 - 4(1)(-54))) / (2 * 1)
Simplifying further:
x = (3 ± sqrt(9 + 216)) / 2
x = (3 ± sqrt(225)) / 2
x = (3 ± 15) / 2
This gives us two possible values for x:
x = (3 + 15) / 2 = 18 / 2 = 9
x = (3 - 15) / 2 = -12 / 2 = -6
Since the length of a rectangle cannot be negative, we discard the negative value.
Therefore, the length of the rectangle is 9 cm.
To find the width, we substitute the value of x into the equation we derived earlier:
Width = x - 3
Width = 9 - 3
Width = 6
Hence, the width of the rectangle is 6 cm, and the length is 9 cm.