2. The length of a rectangle is four meters less than twice its width.
If the area of the rectangle is 96 m^2, what is the length and the width?
(3 points)
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An expression for the length of the rectangle in terms of the width would be Response area
The formula for the area of a rectangle is Response area
Using trial and error, if the area is 96 m^2, then the length and width are
Let's assume the width of the rectangle is x meters.
According to the given information, the length of the rectangle is four meters less than twice its width.
So the length can be expressed as 2x - 4 meters.
The formula for the area of a rectangle is length multiplied by width.
Therefore, the area of the rectangle can be expressed as (2x - 4) * x = 96.
Now we can solve this equation for x:
2x^2 - 4x = 96
2x^2 - 4x - 96 = 0
Dividing the equation by 2, we get:
x^2 - 2x - 48 = 0
Now we can solve this quadratic equation by factoring:
(x - 8)(x + 6) = 0
So x = 8 or x = -6.
Since the width of a rectangle cannot be negative, we discard the negative solution.
Therefore, the width of the rectangle is 8 meters.
Substituting this value back into our expression for the length:
Length = 2x - 4 = 2 * 8 - 4 = 12 meters.
So the length of the rectangle is 12 meters and the width is 8 meters.
Let's solve this step by step.
1. First, let's assume the width of the rectangle. Let's call it "w" (in meters).
2. According to the given information, the length of the rectangle is four meters less than twice its width. So, the length can be expressed as "2w - 4" (in meters).
3. The formula for the area of a rectangle is length multiplied by width. So, the area of the rectangle can be calculated as (2w - 4) * w = 96 m^2.
4. Now, let's solve the equation to find the value of w:
(2w - 4) * w = 96
2w^2 - 4w = 96
2w^2 - 4w - 96 = 0
5. To solve this quadratic equation, we can either factorize it or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = -4, and c = -96.
w = (-(-4) ± √((-4)^2 - 4*2*(-96))) / (2*2)
= (4 ± √(16 + 768)) / 4
= (4 ± √784) / 4
= (4 ± 28) / 4
So, w = (4 + 28)/4 = 32/4 = 8 or w = (4 - 28)/4 = -24/4 = -6.
Since width cannot be negative, we discard the value of w = -6.
6. Now that we have the value of the width, we can calculate the length using the expression we derived earlier:
Length = 2w - 4
= 2 * 8 - 4
= 16 - 4
= 12 meters.
Therefore, the length of the rectangle is 12 meters and the width is 8 meters.