In this problem, we have a rectangular garden where the length (L) is defined as being 3 meters more than twice the width (W). The relationship can be expressed as:
\[ L = 2W + 3 \]
The total perimeter (P) of the garden is given as 60 meters, and the formula for the perimeter of a rectangle is:
\[ P = 2L + 2W \]
So we have:
\[ 2L + 2W = 60 \]
We can simplify this to:
\[ L + W = 30 \]
Now, we can substitute the expression for L (from the earlier equation) into this perimeter equation:
\[ (2W + 3) + W = 30 \] \[ 3W + 3 = 30 \] \[ 3W = 27 \] \[ W = 9 \]
Now we can substitute W back to find L:
\[ L = 2(9) + 3 = 18 + 3 = 21 \]
- Width (W) = 9 meters
- Length (L) = 21 meters
Constraints for viable solutions:
Given that the width and length must be positive measurements in the context of garden dimensions, we consider the following constraints:
- Non-negativity: Both the length and width must be non-negative.
- Positive dimensions: Since we are dealing with physical dimensions, length and width must be positive numbers.
Therefore, in terms of the options provided:
- All positive integers: This is too restrictive because it excludes non-integer dimensions that are still viable in this context.
- All positive integers greater than zero: This is similar to positive integers, also too restrictive.
- All real numbers: This would include negative and zero dimensions, which are not viable.
The most appropriate choice would be recognizing that both dimensions need to be greater than zero. However, since that specific option is not listed but similar options are:
- The closest suitable option you have is All real numbers, but in practice, we only consider positive values for valid garden dimensions.
For a more precise answer, the constraints are that both \( W > 0 \) and \( L > 0 \).
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