The width of a rectangle is 6 cm less than its length. If the area of the rectangle is 135 square cm, find the dimensions. Then find the perimeter of the rectangle.

W = Width

L = Length

A = Area

W = L - 6

A = W * L

135 =( L - 6 ) * L

135 = L ^ 2 - 6 L

The exact solution are :

L = - 9 cm

and

L = 15 cm

The length can't be negative so:

L = 15 cm

W = L - 6

W = 15 - 6 = 9 cm

A = W * L = 15 * 9 = 135 cm ^ 2

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Why did the rectangle go on a diet? Because it wanted to keep its width 6 cm less than its length!

Let's call the length of the rectangle "L". So, the width would be L - 6 cm.

Now, we know that the area of a rectangle is given by Length × Width, which in this case is L × (L - 6).

Setting the area equal to 135 square cm, we have:

L × (L - 6) = 135

Expanding this equation gives us:

L² - 6L - 135 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula:

(L - 15)(L + 9) = 0

So, L - 15 = 0 or L + 9 = 0.

If L - 15 = 0, then L = 15. If L + 9 = 0, then L = -9.

Since length cannot be negative, we discard L = -9.

Therefore, the length of the rectangle is L = 15 cm, and the width is L - 6 = 15 - 6 = 9 cm.

To find the perimeter of the rectangle, we use the formula P = 2L + 2W, where P is the perimeter, L is the length, and W is the width:

P = 2(15) + 2(9)
P = 30 + 18
P = 48 cm

So, the dimensions of the rectangle are 15 cm by 9 cm, and its perimeter is 48 cm.

Let's solve the problem step by step.

Step 1: Let's assign variables to the dimensions of the rectangle.
Let the length of the rectangle be "L" cm.
Therefore, the width of the rectangle would be "L - 6" cm.

Step 2: Use the formula for the area of a rectangle.
The area of a rectangle is given by the formula A = length × width.
So, we can write the equation:
A = L × (L - 6)
Given that the area is 135 square cm, we can substitute this value into the equation:
135 = L × (L - 6)

Step 3: Solve the equation.
To solve this equation, we can expand the brackets and rearrange the terms:
135 = L^2 - 6L
Now, bring 135 to the left side of the equation and set it equal to zero:
L^2 - 6L - 135 = 0

Step 4: Factorize or solve for L.
To solve this quadratic equation, we can either factorize it or use the quadratic formula. Factoring may not be possible in this case, so let's use the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation L^2 - 6L - 135 = 0, the coefficients are:
a = 1, b = -6, and c = -135
Substituting these values into the quadratic formula, we get:
L = (6 ± sqrt((-6)^2 - 4(1)(-135))) / (2(1))

Simplifying this equation will give us two possible values for L.

Step 5: Calculate the possible values for L.
L = (6 ± sqrt(36 + 540)) / 2
L = (6 ± sqrt(576)) / 2
L = (6 ± 24) / 2
L = (6 + 24) / 2 or L = (6 - 24) / 2
L = 30 / 2 or L = -18 / 2
L = 15 or L = -9

Since the length cannot be negative, we discard -9 as a solution.

Step 6: Find the width of the rectangle.
Using the length we found, we can calculate the width of the rectangle.
When L = 15, the width = L - 6 = 15 - 6 = 9 cm.

So, the dimensions of the rectangle are:
Length = 15 cm
Width = 9 cm

Step 7: Calculate the perimeter of the rectangle.
The perimeter of a rectangle is given by the formula:
Perimeter = 2 × (length + width)
Substituting the values we found, we get:
Perimeter = 2 × (15 + 9)
Perimeter = 2 × 24
Perimeter = 48 cm

Therefore, the dimensions of the rectangle are 15 cm by 9 cm, and its perimeter is 48 cm.

To find the dimensions of the rectangle, we need to set up an equation based on the given information.

Let's assume that the length of the rectangle is "x" cm.
According to the given information, the width of the rectangle is 6 cm less than its length, so the width would be (x - 6) cm.

The area of a rectangle is calculated by multiplying its length by its width. In this case, we know the area is 135 square cm, so we can set up the following equation:

Length × Width = 135

Substituting the expressions for length and width, we have:

x × (x - 6) = 135

Expanding the equation, we get:

x^2 - 6x = 135

Rearranging the equation into standard form, we obtain the quadratic equation:

x^2 - 6x - 135 = 0

Now, to solve this quadratic equation, we can either factor it or use the quadratic formula.

To solve the equation by factoring, we need to find two numbers whose product is -135 and whose sum is -6. The numbers are -15 and 9, so we can rewrite the equation as:

(x - 15)(x + 9) = 0

Setting each factor equal to zero, we have two possible solutions for x:

x - 15 = 0 -> x = 15

or

x + 9 = 0 -> x = -9

Since the length cannot be negative, we disregard the second solution.

Therefore, the length of the rectangle is 15 cm.

To find the width, we can substitute the value of x back into the expression (x - 6):

Width = x - 6 = 15 - 6 = 9 cm

So, the dimensions of the rectangle are: length = 15 cm and width = 9 cm.

Now, let's calculate the perimeter of the rectangle. The perimeter can be found by adding up the lengths of all four sides of the rectangle.

Perimeter = 2(length + width)

Substituting the values we found, we have:

Perimeter = 2(15 + 9) = 2(24) = 48 cm

Therefore, the perimeter of the rectangle is 48 cm.

perimeter of a rectangle with length 6.6m and width 4.5