A pet boarding facility wants to build two adjacent pens of equal size. They are using the length of the dog kennel as one side of the two pens. They have 324 feet of fencing to build the pens. What width should he make the pens to maximize the total area of the pens?

fence=3w+l=324

area=lw=(324-3w)w
zeroes at w=0, w=325/3

so max should be 1/2 way between the two zeroes, or max w=324/6=54
l=324-3(54)=you do it.

A square pens will give you the maximum area. To build 2 adjacent square pens with one common side, you will need 6 equal lenghts of fence.

324/6 = 54
Area of each square pens = 54×54=2916 sq ft

I forgot to mention that the lenght of the dog kennel is the common side and therefore need not taken from the available fencing.

To maximize the total area of the pens, we can follow these steps:

Step 1: Understand the problem:
- The pet boarding facility wants to build two adjacent pens of equal size.
- The length of the dog kennel will be one side of the two pens.
- The total length of fencing available is 324 feet.
- We need to determine the width of the pens that will maximize the total area.

Step 2: Define the variables:
- Let's assume the length of the dog kennel as 'L'.
- The width of each pen will be 'W'.
- The total length of fencing is given as 324 feet.

Step 3: Formulate the equation for the total area:
- The area 'A' of each pen is given by A = L * W.
- Since we have two pens, the total area 'T' of the pens is T = 2 * A = 2 * (L * W) = 2LW.

Step 4: Express the perimeter equation in terms of a single variable:
- The perimeter P of the pens is given by P = 2L + 3W.
- However, we are given that the total length of fencing available is 324 feet, so 2L + 3W = 324.

Step 5: Solve the perimeter equation for one variable:
- Rearrange the perimeter equation to express one variable in terms of the other. We can solve for L in terms of W:
2L + 3W = 324
2L = 324 - 3W
L = (324 - 3W) / 2

Step 6: Substitute the value of L into the equation for the total area:
- Now we have the total area equation in terms of one variable (W):
T = 2LW

Step 7: Maximize the total area:
- To maximize the total area, we need to find the value of W that maximizes T.
- We can achieve this by taking the derivative of T with respect to W, setting it equal to zero, and solving for W.
- However, since this is a quadratic equation, we can instead use the vertex formula to find the maximum value.

Step 8: Apply the vertex formula:
- The formula to find the vertex of a quadratic equation in the form ax^2 + bx + c is given by x = -b / (2a).

- In our case, the quadratic equation is T = 2LW, so a = 2L and b = 0.
- Substituting these values into the vertex formula, we get W = -0 / (2 * 2L), which simplifies to W = 0.
- This means that the vertex of the quadratic equation occurs when W is 0.

Step 9: Interpret the result:
- Since a width of 0 would result in a single pen, the maximum area occurs when the width of each pen is as large as possible.
- In practical terms, this means that the pens would be as wide as the available fencing allows, effectively creating one large pen instead of two adjacent pens.
- Therefore, to maximize the total area, the width of the pens should be as wide as possible, which is equal to the available fencing length of 324 feet.

In conclusion, the pens should have a width equal to the available fencing length of 324 feet in order to maximize the total area of the pens.