The area of a rectangle is 189. The width is 2 meter longer than 1/3 the length. find the length and the width

width=x, length=y

x = y/3 + 2

xy = 189
(y/3 + 2)y = 189
y^2/3 + 2y - 189 = 0
y^2 + 6y - 567 = 0
(y-21)(y+27)

no negative sides, so y=21 and x=9. xy=189

Let's denote the length of the rectangle as "L" and the width as "W". We have two pieces of information given in the problem:

1) The area of the rectangle is 189:
We know that the formula for the area of a rectangle is A = length * width. Substituting the given values, we get:
189 = L * W

2) The width is 2 meters longer than 1/3 of the length:
Expressing this in an equation, we have:
W = (1/3) * L + 2

Now we can solve the system of equations to find the length and width of the rectangle. Let's substitute the value of W from equation 2 into equation 1:

189 = L * ((1/3) * L + 2)

To simplify the equation, multiply L inside the parentheses:

189 = (1/3) * L^2 + 2L

Multiplying both sides of the equation by 3 to get rid of the fraction:

567 = L^2 + 6L

Rearranging the equation to set it equal to zero:

L^2 + 6L - 567 = 0

Now we can factorize or use the quadratic formula to find the solutions for L. Factoring the equation, we have:

(L + 27)(L - 21) = 0

Setting each factor equal to zero and solving for L:
L + 27 = 0 --> L = -27 (disregard negative value in this case)
L - 21 = 0 --> L = 21

We found that the length of the rectangle is 21 meters. Now we can substitute this value back into equation 2 to find the width:

W = (1/3) * 21 + 2 = 7 + 2 = 9

Therefore, the length of the rectangle is 21 meters and the width is 9 meters.

To find the length and width of the rectangle, we can set up equations based on the given information.

Let's assume the length of the rectangle is "L" meters.
According to the given information, the width of the rectangle is 2 meters longer than one-third of the length.
So, the width can be expressed as 1/3L + 2 meters.

The area of a rectangle is found by multiplying its length and width.
We know that the area is 189 square meters, so we can set up the following equation:
Length × Width = 189

Plugging in the expressions for length and width, we have:
L × (1/3L + 2) = 189

Now, let's solve this equation to find the value of L (the length).

Multiplying L by both terms inside the parentheses:
(1/3) L^2 + 2L = 189

To simplify the equation, let's multiply everything by 3 to get rid of the fraction:
L^2 + 6L = 567

Rearranging the equation in standard form:
L^2 + 6L - 567 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula. Using the quadratic formula:
L = (-6 ± √(6^2 - 4(1)(-567))) / (2(1))

Simplifying the expression under the square root:
L = (-6 ± √(36 + 2268)) / 2
L = (-6 ± √(2304)) / 2
L = (-6 ± 48) / 2

Now we have two possible values for L:
L₁ = (-6 + 48) / 2 = 42 / 2 = 21
L₂ = (-6 - 48) / 2 = -54 / 2 = -27

Since the length cannot be negative, we ignore the second solution. Therefore, the length of the rectangle is 21 meters.

Now, let's find the width. Plugging in the value of L into the width expression:
Width = 1/3L + 2
Width = 1/3(21) + 2
Width = 7 + 2
Width = 9

Thus, the length of the rectangle is 21 meters, and the width is 9 meters.

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