Math  /  Algebra

Question1. The length of a rectangle is 3 meters less than twice its width. The area of the rectangle is 104 m2104 \mathrm{~m}^{2}. Find the dimensions of the rectangle.

Studdy Solution

STEP 1

1. The length of the rectangle is 3 meters less than twice its width.
2. The area of the rectangle is 104m2104 \, \text{m}^2.
3. We need to find the dimensions of the rectangle, i.e., its length and width.

STEP 2

1. Define variables for the dimensions of the rectangle.
2. Write equations based on the problem description.
3. Solve the equations to find the dimensions of the rectangle.

STEP 3

Define variables for the dimensions of the rectangle.
Let w w be the width of the rectangle in meters. Let l l be the length of the rectangle in meters.

STEP 4

Write equations based on the problem description.
From the problem, we know: - The length l l is 3 meters less than twice the width: l=2w3 l = 2w - 3
- The area of the rectangle is given by the formula Area=length×width \text{Area} = \text{length} \times \text{width} , so: l×w=104 l \times w = 104
Substitute the expression for l l from the first equation into the area equation: (2w3)×w=104 (2w - 3) \times w = 104

STEP 5

Solve the equation to find the dimensions of the rectangle.
Expand the equation: 2w23w=104 2w^2 - 3w = 104
Rearrange the equation to form a standard quadratic equation: 2w23w104=0 2w^2 - 3w - 104 = 0

STEP 6

Solve the quadratic equation using the quadratic formula: The quadratic formula is given by: w=b±b24ac2a w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
For our equation 2w23w104=0 2w^2 - 3w - 104 = 0 , a=2 a = 2 , b=3 b = -3 , and c=104 c = -104 .
Calculate the discriminant: b24ac=(3)24×2×(104) b^2 - 4ac = (-3)^2 - 4 \times 2 \times (-104) =9+832 = 9 + 832 =841 = 841
Since the discriminant is positive, there are two real solutions.
Calculate the solutions: w=(3)±8412×2 w = \frac{-(-3) \pm \sqrt{841}}{2 \times 2} w=3±294 w = \frac{3 \pm 29}{4}
Calculate the possible values for w w : w=3+294=324=8 w = \frac{3 + 29}{4} = \frac{32}{4} = 8 w=3294=264=6.5 w = \frac{3 - 29}{4} = \frac{-26}{4} = -6.5
Since width cannot be negative, w=8 w = 8 .

STEP 7

Find the corresponding length using the width.
Substitute w=8 w = 8 back into the equation for l l : l=2w3 l = 2w - 3 l=2(8)3 l = 2(8) - 3 l=163 l = 16 - 3 l=13 l = 13
The dimensions of the rectangle are: Width = 8m 8 \, \text{m} Length = 13m 13 \, \text{m}

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