You have 20 feet of fencing material to make a pen for your dog. The design of the pen is given as follows.
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.
To find the perimeter of the dog pen in terms of m and n, we need to add up the lengths of all four sides of the rectangle. In a rectangle, opposite sides are equal in length, so the perimeter is given by:
Perimeter = 2m + 2n
For each set of dimensions, let's substitute the values of m and n into the formula:
1) m = 3 feet, n = 2 feet:
Perimeter = 2(3) + 2(2) = 6 + 4 = 10 feet
2) m = 5.5 feet, n = 3 feet:
Perimeter = 2(5.5) + 2(3) = 11 + 6 = 17 feet
3) m = 7.5 feet, n = 5.5 feet:
Perimeter = 2(7.5) + 2(5.5) = 15 + 11 = 26 feet
Now, let's see if we can create a dog pen with the given dimensions using the available 20 feet of fencing material:
1) m = 3 feet, n = 2 feet:
Perimeter = 10 feet
Since the perimeter is less than the available 20 feet of fencing material, we can create a dog pen with these dimensions.
2) m = 5.5 feet, n = 3 feet:
Perimeter = 17 feet
The perimeter exceeds the available 20 feet of fencing material, so we cannot create a dog pen with these dimensions.
3) m = 7.5 feet, n = 5.5 feet:
Perimeter = 26 feet
Again, the perimeter exceeds the available 20 feet of fencing material, so we cannot create a dog pen with these dimensions.
We need to find the dimensions that have a perimeter less than or equal to 20 feet. Possible dimensions can be found by substituting different values of m and n into the perimeter formula and checking if the result is less than or equal to 20.
For example, let's find a possible dimension using m = 4 feet:
Perimeter = 2(4) + 2n = 8 + 2n
Setting the perimeter to be equal to 20:
8 + 2n = 20
2n = 20 - 8
2n = 12
n = 6
So, a possible dimension could be m = 4 feet, n = 6 feet.
Similarly, we can continue to find other possible dimensions by substituting values of m and solving the perimeter equation.
If we want to maximize the size of the dog pen while using all of the available fencing material, we need to find the dimensions that have the largest possible perimeter, which is 20 feet. By setting the perimeter equal to 20 and solving for n, we can find the corresponding value of m.