The length, l, of a rectangle is 2 feet less than three times the width, w, of the rectangle. The area of the rectangle is 481 square feet. Find the length and width of the rectangle.

how do i work this problem out?

To solve this problem, we can use the information given to create two equations.

Let's start by assigning variables to the unknowns:

Let the width of the rectangle be w in feet.
Let the length of the rectangle be l in feet.

From the given information, we can create two equations:

Equation 1: l = 3w - 2 (Length is 2 feet less than three times the width)
Equation 2: l * w = 481 (Area of the rectangle is 481 square feet)

Now we can solve the system of equations by substituting Equation 1 into Equation 2:

(3w - 2) * w = 481

Expanding the equation:

3w^2 - 2w = 481

Rearranging the equation to form a quadratic equation:

3w^2 - 2w - 481 = 0

Next, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / 2a

Using the quadratic formula with a = 3, b = -2, and c = -481, we get:

w = (2 ± √((-2)^2 - 4 * 3 * (-481))) / (2 * 3)
w = (2 ± √(4 + 5784)) / 6
w = (2 ± √5788) / 6

Calculating the two possible values for w:

w ≈ (2 + √5788) / 6 ≈ 11.15
w ≈ (2 - √5788) / 6 ≈ -9.81 (this value is not meaningful in this context, as width cannot be negative)

So, the width of the rectangle is approximately 11.15 feet.

Plugging this value into Equation 1 to find the length:

l = 3w - 2
l ≈ 3(11.15) - 2
l ≈ 33.45 - 2
l ≈ 31.45

So, the length of the rectangle is approximately 31.45 feet.

Therefore, the length and width of the rectangle are approximately 31.45 feet and 11.15 feet, respectively.

To solve this problem, let's break it down step by step.

Step 1: Set up the given information as equations.
We are given two pieces of information: the length (l) is 2 feet less than three times the width (w), and the area of the rectangle is 481 square feet. We can write these as equations:
l = 3w - 2 (equation 1)
lw = 481 (equation 2)

Step 2: Substitute equation 1 into equation 2.
Since we know that l = 3w - 2, we can substitute this into equation 2:
(3w - 2)w = 481

Step 3: Simplify the equation.
Distribute the w into the parentheses:
3w^2 - 2w = 481

Step 4: Rearrange the equation.
Move the 481 to the other side of the equation to set it equal to zero:
3w^2 - 2w - 481 = 0

Step 5: Solve the quadratic equation.
To solve this quadratic equation, you can either factor it or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 3, b = -2, and c = -481. Plugging in these values, we get:
w = (-(-2) ± √((-2)^2 - 4(3)(-481))) / (2(3))
w = (2 ± √(4 + 5784)) / 6
w = (2 ± √5788) / 6
w ≈ (2 ± 76.08) / 6

Simplifying this expression, we get two possible values for w:
w ≈ (2 + 76.08) / 6 ≈ 78.08 / 6 ≈ 13.01
w ≈ (2 - 76.08) / 6 ≈ -74.08 / 6 ≈ -12.35

Since the width cannot be negative, we discard the negative value and take the positive value of w as the width of the rectangle.

Step 6: Find the length.
Now that we have the value of w, we can substitute it back into equation 1 to find the length:
l = 3w - 2
l = 3(13.01) - 2
l ≈ 39.03 - 2
l ≈ 37.03

So, the length of the rectangle is approximately 37.03 feet and the width is approximately 13.01 feet.

length = 3w-2

w is width

area= w(3w-2)