The length of a rectangle is 5 centimeters less than three times it's width . It's area is 12 square centimeters. Find the dimensions of the rectangle
w(3w-5) = 12
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To find the dimensions of the rectangle, we need to set up an equation using the given information.
Let's assume the width of the rectangle is "w" centimeters.
According to the problem, the length of the rectangle is 5 centimeters less than three times its width. So, the length can be expressed as (3w - 5) centimeters.
The area of a rectangle is given by the formula A = length * width.
In this case, the area is given as 12 square centimeters.
So, we can set up the equation as:
A = (3w - 5) * w
12 = (3w - 5) * w
Now, we can solve this equation to find the value of "w" (width), and once we have that, we can find the length as well.
Expanding the equation, we get:
12 = 3w^2 - 5w
Rearranging the terms, we get a quadratic equation:
3w^2 - 5w - 12 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula, which states that for an equation in the form ax^2 + bx + c = 0, the solutions for "x" are given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -5, and c = -12. Plugging in these values into the quadratic formula, we get:
w = (-(-5) ± sqrt((-5)^2 - 4 * 3 * (-12))) / (2 * 3)
w = (5 ± sqrt(25 + 144)) / 6
w = (5 ± sqrt(169)) / 6
w = (5 ± 13) / 6
This gives us two possible values for the width, w:
w = (5 + 13) / 6 or w = (5 - 13) / 6
Simplifying, we get:
w = 18 / 6 or w = -8 / 6
w = 3 or w = -4/3
Since width cannot be negative, we discard the negative value.
Therefore, the width of the rectangle is 3 centimeters.
To find the length, we substitute the value of the width back into the expression for the length:
Length = 3w - 5
Length = 3(3) - 5
Length = 9 - 5
Length = 4 centimeters
So, the dimensions of the rectangle are: width = 3 centimeters and length = 4 centimeters.