Two less than three times the width of a rectangle is equal to the length. The area of the rectangle is 65 square ft. What is the length of the rectangle?
just plug in the data and solve for w. Then you can get the length.
w(3w-2) = 65
Of course, it also helps to know that 5*13 = 65 ...
just sayin'
To solve this problem, let's follow these steps:
Step 1: Assign variables
Let's assign variables to the unknown values.
Let width = w
Let length = l
Step 2: Set up the equation
According to the problem, "Two less than three times the width of a rectangle is equal to the length." We can express this in an equation: "3w - 2 = l"
Step 3: Find the width of the rectangle
We know that the area of the rectangle is 65 square feet. The formula for the area of a rectangle is: "Area = length * width". Plugging in the values, we get: "65 = l * w".
Step 4: Solve the equation
We have a system of equations:
Equation 1: 3w - 2 = l
Equation 2: 65 = l * w
To solve the system, we can use substitution. We'll substitute Equation 1 into Equation 2:
65 = (3w - 2) * w
Expanding this equation:
65 = 3w^2 - 2w
Rearranging and setting it equal to zero:
3w^2 - 2w - 65 = 0
Step 5: Solve the quadratic equation
To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -2, and c = -65:
w = (-(-2) ± √((-2)^2 - 4(3)(-65))) / (2(3))
Simplifying inside the square root:
w = (2 ± √(4 + 780)) / 6
w = (2 ± √784) / 6
w = (2 ± 28) / 6
Solving for w:
w = (2 + 28) / 6 or w = (2 - 28) / 6
w = 30 / 6 or w = -26 / 6
Simplifying:
w = 5 or w ≈ -4.33
Step 6: Determine the length
Since the width of a rectangle cannot be negative, we take the positive value of w. So, the width is 5 feet.
Now we can plug it back into any of the equations to find the length. Using Equation 1:
3w - 2 = l
Substituting the value of w as 5:
3(5) - 2 = l
Simplifying:
15 - 2 = l
l = 13
Therefore, the length of the rectangle is 13 feet.
To find the length of the rectangle, we need to rewrite the information given in the question as an equation.
Let's say the width of the rectangle is 'w'.
According to the information given, "Two less than three times the width of a rectangle is equal to the length." can be expressed as:
3w - 2 = length
The area of the rectangle is given as 65 square ft. Since the formula to find the area of a rectangle is Length x Width, we can write:
w * length = 65
Now, we can substitute the value of length from the first equation into the second equation:
w * (3w - 2) = 65
Simplifying the equation:
3w^2 - 2w - 65 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. I'll use the quadratic formula here:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 3, b = -2, and c = -65. Plugging in these values, we get:
w = (-(-2) ± √((-2)^2 - 4 * 3 * -65)) / (2 * 3)
w = (2 ± √(4 + 780)) / 6
w = (2 ± √784) / 6
w = (2 ± 28) / 6
Simplifying further, we have:
w = (2 + 28) / 6 = 30 / 6 = 5
w = (2 - 28) / 6 = -26 / 6 = -13/3
Since the width of a rectangle cannot be negative, we discard the -13/3 value.
Therefore, the width of the rectangle is 5 ft.
Now, substituting this value into the first equation to find the length:
3w - 2 = length
3 * 5 - 2 = length
15 - 2 = length
13 = length
Therefore, the length of the rectangle is 13 ft.