the area of a rectangle is 65yd^2 the length is 3yd less than double the width. Find the dimensions
L = (2 W - 3)
L * W = 65
(2 W - 3) W = 65
2 W^2 - 3 W - 65 = 0
( 2 W - 13) ( W + 5) = 0
-5 will just not do so W = 13/2 = 6.5
then L = 10
To find the dimensions of the rectangle, we can use the given information that the area is 65 yd² and the length is 3 yd less than double the width.
Let's break down the information:
1. The area of a rectangle is given by the formula: Area = Length × Width.
2. The given area of the rectangle is 65 yd².
From the second piece of information, we can establish an equation:
Length × Width = 65
Now, let's use the second part of the given information: the length is 3 yd less than double the width. Mathematically, we can express this as:
Length = 2 × Width - 3
Substituting this expression for length in terms of width into the area equation:
(2 × Width - 3) × Width = 65
Now, we have a quadratic equation. Expanding and rearranging it:
2 × Width² - 3 × Width - 65 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Since factoring is not possible for this equation, we'll use the quadratic formula:
Width = (-b ± sqrt(b² - 4ac)) / (2a)
For our equation, a = 2, b = -3, and c = -65. Substituting these into the quadratic formula:
Width = (3 ± sqrt((-3)² - 4 × 2 × (-65))) / (2 × 2)
Simplifying:
Width = (3 ± sqrt(9 + 520)) / 4
Width = (3 ± sqrt(529)) / 4
Width = (3 ± 23) / 4
Now, we have two possible values for the width. Let's consider these separately:
1. If Width = (3 + 23)/4 = 26/4 = 6.5 yd
Substituting this value into the equation for the length:
Length = 2 × 6.5 - 3 = 13 - 3 = 10 yd
So, one possible set of dimensions is Width = 6.5 yd and Length = 10 yd.
2. If Width = (3 - 23)/4 = -20/4 = -5 yd (Since dimensions cannot be negative, we discard this value.)
Therefore, the dimensions of the rectangle are Width = 6.5 yd and Length = 10 yd.