To set up the constraints for the rectangular vegetable garden, we start by defining the variables involved:
- Let \( w \) be the width of the garden (in meters).
- The length \( l \) of the garden is given as \( l = 2w + 3 \) (because the length is always 3 meters more than twice the width).
Next, we know the perimeter \( P \) of the rectangle is given by the formula:
\[ P = 2l + 2w \]
From the problem, we are told that the total perimeter must be 60 meters. Therefore, we have:
\[ 2l + 2w = 60 \]
Substituting for \( l \) from the earlier equation:
\[ 2(2w + 3) + 2w = 60 \]
Now, we simplify this equation:
\[ 4w + 6 + 2w = 60 \]
Combining like terms gives us:
\[ 6w + 6 = 60 \]
Subtracting 6 from both sides results in:
\[ 6w = 54 \]
Dividing by 6, we find:
\[ w = 9 \]
Now substituting \( w \) back into the equation for \( l \):
\[ l = 2(9) + 3 = 18 + 3 = 21 \]
So, the dimensions of the garden are \( w = 9 \) meters and \( l = 21 \) meters.
Constraints Summary
Now to list the constraints for viable solutions:
- Non-negativity Constraint: Both the width and length must be positive numbers.
- \( w > 0 \)
- \( l > 0 \)
Since \( l = 2w + 3 \), this automatically ensures \( w > 0 \) implies \( l > 3 \).
- Perimeter Constraint: The total perimeter must equal 60 meters.
- \( 2l + 2w = 60 \)
Given these conditions, we summarize:
- \( w > 0 \)
- \( l = 2w + 3 \)
- \( 2l + 2w = 60 \)
This gives a clear understanding of the relationships and restrictions of the garden dimensions.
Also note the additional implicit constraint derived from the equation:
- As perimeter must be maintained, if \( w \) is increased, \( l \) must also adjust based on the overall set relationship, but it must not exceed the physical limits of the perimeter constraint.
This situation allows only the positive dimensions that satisfy the perimeter condition. Hence, our derived solution \( (w, l) = (9, 21) \) respects all necessary constraints for practical construction.
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