The length of a rectangle is
3yd less than twice the width, and the area of the rectangle is 65ydsquare. Find the dimensions of the rectangle.
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To find the dimensions of the rectangle, let's define the width as 'w' and the length as 'l'.
From the given information, we can set up two equations.
Equation 1: "The length of a rectangle is 3yd less than twice the width."
This can be expressed as: l = 2w - 3
Equation 2: "The area of the rectangle is 65ydsquare."
The formula for the area of a rectangle is: Area = length * width, which in this case is: 65 = l * w
Now we have a system of two equations:
Equation 1: l = 2w - 3
Equation 2: 65 = l * w
To solve this system, we can substitute the value of 'l' from Equation 1 into Equation 2. So, we have:
65 = (2w - 3) * w
Expanding the equation gives us:
65 = 2w^2 - 3w
Rearranging:
2w^2 - 3w - 65 = 0
Now we have a quadratic equation, and to solve it we can use factorization, completing the square, or the quadratic formula. Let's use the quadratic formula:
w = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the values from the quadratic equation:
w = (-(-3) ± sqrt((-3)^2 - 4 * 2 * (-65))) / (2 * 2)
Simplifying:
w = (3 ± sqrt(9 + 520)) / 4
w = (3 ± sqrt(529)) / 4
w = (3 ± 23) / 4
Simplifying further:
Case 1: w = (3 + 23) / 4 = 26 / 4 = 6.5
Case 2: w = (3 - 23) / 4 = -20 / 4 = -5
Since the width cannot be negative, we discard the negative value.
Therefore, the width of the rectangle is 6.5 yards.
Now, we can substitute the value of 'w' into Equation 1 to find the length:
l = 2w - 3
l = 2 * 6.5 - 3
l = 13 - 3
l = 10
Therefore, the length of the rectangle is 10 yards.
In conclusion, the dimensions of the rectangle are: width = 6.5 yards and length = 10 yards.