To find the perimeter of a dog pen designed such that it has width \( m \) and height \( n \), we use the formula for the perimeter of a rectangle:
\[ P = 2(m + n) \]
Given that you have 20 feet of fencing material, we set the perimeter equal to 20 feet:
\[ 2(m + n) = 20 \]
Dividing both sides by 2 gives:
\[ m + n = 10 \]
Now, let's check if the proposed dimensions can be accommodated within the available fencing material.
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For \( m = 3 \) feet and \( n = 2 \) feet: \[ m + n = 3 + 2 = 5 \quad (\text{less than } 10) \] The perimeter is \( 2(3 + 2) = 10 \) feet, so yes, this can be created but will not use all available fencing.
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For \( m = 5.5 \) feet and \( n = 3 \) feet: \[ m + n = 5.5 + 3 = 8.5 \quad (\text{less than } 10) \] The perimeter is \( 2(5.5 + 3) = 17 \) feet, so yes, this can also be created but will not use all available fencing.
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For \( m = 7.5 \) feet and \( n = 5.5 \) feet: \[ m + n = 7.5 + 5.5 = 13 \quad (\text{greater than } 10) \] The perimeter is \( 2(7.5 + 5.5) = 26 \) feet, so this cannot be created as it exceeds the available fencing.
Possible dimensions of the dog pen:
The possible dimensions based on the requirement \( m + n = 10 \) are any pairs of \( (m, n) \) that satisfy this equation. For instance:
- \( m = 0 \), \( n = 10 \)
- \( m = 1 \), \( n = 9 \)
- \( m = 2 \), \( n = 8 \)
- \( m = 3 \), \( n = 7 \)
- \( m = 4 \), \( n = 6 \)
- \( m = 5 \), \( n = 5 \)
- \( m = 6 \), \( n = 4 \)
- \( m = 7 \), \( n = 3 \)
- \( m = 8 \), \( n = 2 \)
- \( m = 9 \), \( n = 1 \)
- \( m = 10 \), \( n = 0 \)
Among the dimensions listed, we should choose dimensions that make the pen functional. Reasoning suggests that dimensions providing more area for the dog would be preferable. Hence, among the dimensions possible, one of the options maximizing area could be \( m = 5 \) feet and \( n = 5 \) feet, which would give the most balanced area of \( 5 \times 5 = 25 \) square feet.