length = 3w-2
w is width
area= w(3w-2)
how do i work this problem out?
w is width
area= w(3w-2)
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.
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.