You have 20 feet of fencing material to make a pen for your dog. The design of the pen is given as follows.

Question

An illustration shows a rectangle named as Dog Pen with length as lower m and width as lower n.
Find the perimeter of the dog pen in terms of  m  and  n .
Will you be able to create a dog pen with the following dimensions? Explain your answers.  
m=3 feet by n=2 feet
 
m=5.5 feet by n=3 feet
 
m=7.5 feet by n=5.5 feet
 
List the possible dimensions of the dog pen that you can make out of the available fencing material.  
Among the dimensions you listed, which should you use to make the dog pen? Explain your reasoning.

GPT-4o mini · Answer

To find the perimeter of the dog pen in terms of \( m \) (length) and \( n \) (width), we use the formula for the perimeter of a rectangle:

\[ \text{Perimeter} = 2m + 2n \]

Since you have 20 feet of fencing material, we set the equation for the perimeter equal to 20:

\[ 2m + 2n = 20
\]

Dividing both sides by 2 gives:

\[ m + n = 10 \]

Now, let's evaluate whether the following dimensions can be created using the 20 feet of fencing.

For \( m = 3 \) feet and \( n = 2 \) feet: \[ m + n = 3 + 2 = 5 \quad (\text{less than } 10) \]

This can be created, but it only accounts for 10 feet of fencing, which is not using the full amount available.
For \( m = 5.5 \) feet and \( n = 3 \) feet: \[ m + n = 5.5 + 3 = 8.5 \quad (\text{less than } 10) \]

This can also be created, but it uses only 17 feet of fencing.
For \( m = 7.5 \) feet and \( n = 5.5 \) feet: \[ m + n = 7.5 + 5.5 = 13 \quad (\text{greater than } 10) \]

This cannot be created because it exceeds the available 20 feet of fencing.

Possible Dimensions:

The only conditions on \( m \) and \( n \) are that they must satisfy \( m + n = 10 \) and both \( m \) and \( n \) must be greater than 0. This gives a range of possible dimensions, such as:

\( m = 1 \) feet, \( n = 9 \) feet
\( m = 2 \) feet, \( n = 8 \) feet
\( m = 3 \) feet, \( n = 7 \) feet
\( m = 4 \) feet, \( n = 6 \) feet
\( m = 5 \) feet, \( n = 5 \) feet
\( m = 6 \) feet, \( n = 4 \) feet
\( m = 7 \) feet, \( n = 3 \) feet
\( m = 8 \) feet, \( n = 2 \) feet
\( m = 9 \) feet, \( n = 1 \) feet

Recommended Dimensions:

When choosing dimensions for your dog pen, you might want to consider the following:

Space for the Dog: Larger dimensions provide more space for your dog to move around comfortably.
Material Usage: Since you have 20 feet available, you should try to utilize it efficiently while adhering to the space needs of your dog.

Based on the above reasoning, the dimensions of \( m = 5 \) feet and \( n = 5 \) feet give a square which maximizes the area in a given perimeter and offers a balanced space for your dog to play. This dimension uses the entire 20 feet of fencing material, providing sufficient area for comfort.