The length of a rectangle is three centimeters less than the width. If the area of the rectangle is 54cm^2 , find the dimensions of the rectangle
width ---- x cm
then length = x-3 xm
x(x-3) = 54
x^2 - 3x - 54 = 0
(x-9)(x+6) = 0
x = 9 or x = -6, but the width couldn't be negative.
the rectangle is 9 cm by 6 cm
or, simply think this:
what 2 numbers multiply to get 54 and have a difference of 3 ??
Let's assign a variable to represent the width of the rectangle. Let's say the width is x cm.
According to the given information, the length of the rectangle is three centimeters less than the width. So, the length would be (x - 3) cm.
The area of a rectangle can be calculated by multiplying the length and the width. We know that the area is 54 cm^2.
Therefore, the equation to represent the area of the rectangle is:
(x - 3) * x = 54
Simplifying the equation:
x² - 3x = 54
Rearranging the equation into a quadratic equation:
x² - 3x - 54 = 0
Now we need to factorize this quadratic equation:
(x - 9)(x + 6) = 0
Setting each factor equal to zero:
x - 9 = 0 or x + 6 = 0
Solving for x in each equation:
x = 9 or x = -6
Since the width cannot be negative, we discard the value -6.
Therefore, the width of the rectangle is 9 cm, and the length is (9 - 3) = 6 cm.
So, the dimensions of the rectangle are 9 cm (width) and 6 cm (length).
To find the dimensions of the rectangle, we can set up a system of equations based on the given information.
Let's assume that the width of the rectangle is "x" centimeters.
According to the problem, the length of the rectangle is three centimeters less than the width. So, the length can be represented as (x - 3) centimeters.
The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is 54cm². So we have:
54cm² = (x - 3)cm × xcm.
Now, we can solve this equation to find the value of "x" (the width) and then calculate the length of the rectangle.
Expanding the equation, we have:
54cm² = x² - 3x.
Rearranging the equation to bring all terms to one side, we get:
x² - 3x - 54cm² = 0.
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
Factoring:
We need to find two numbers that multiply to give -54 but add up to -3. After factoring, we get:
(x - 9)(x + 6) = 0.
Applying the zero product property, we have two possibilities:
1) x - 9 = 0, which gives x = 9.
2) x + 6 = 0, which gives x = -6.
Since length and width cannot be negative, we discard the solution x = -6.
Therefore, the width of the rectangle is 9 cm.
To calculate the length, we substitute this value back into the formula (length = x - 3):
Length = 9cm - 3cm = 6cm.
So the dimensions of the rectangle are width = 9 cm and length = 6 cm.