Marcus has 72 feet of fencing. He wants to build a rectangular pen with the largest possible area. What should the dimensions of the rectangular pen be to maximize the area?

18x18

a square has the largest area for a given perimeter. How far do you get in solving this algebraically?

554

To maximize the area of the rectangular pen, Marcus should build a square-shaped pen.

Let's assume the dimensions of the square-shaped pen are "x" feet by "x" feet.

The perimeter of a square is equal to 4 times one of its sides. Therefore, the equation for the perimeter of the pen can be written as:

Perimeter = 4x (since all sides are equal in a square).

Given that Marcus has 72 feet of fencing, we can set up the equation:

4x = 72

Now, let's solve for "x":

Divide both sides of the equation by 4:
4x / 4 = 72 / 4
x = 18

So, the side length for the square-shaped pen should be 18 feet.

Therefore, the dimensions of the rectangular pen should be 18 feet by 18 feet to maximize the area.

To find the dimensions of the rectangular pen that will give us the largest possible area, we can follow these steps:

1. Let's assume the length of the rectangular pen is "L" and the width is "W."

2. We know that the perimeter of a rectangle is given by the formula: 2L + 2W. In this case, the perimeter equals 72 feet.

So, we can write the equation: 2L + 2W = 72.

3. Since we want to maximize the area, we need to express the area of the rectangle in terms of a single variable.

The formula for the area of a rectangle is: A = L * W.

4. We can solve the equation from step 2 for one variable. Let's solve it for W:

2L + 2W = 72.
2W = 72 - 2L.
W = 36 - L.

5. Now we can substitute the value of W from step 4 into the area formula from step 3:

A = L * (36 - L).
A = 36L - L^2.

6. To find the maximum area, we can take the derivative of the area equation and set it equal to zero:

dA/dL = 36 - 2L = 0.

Solving for L, we find L = 18.

7. Now that we have a value for L, we can substitute it back into the equation from step 4 to find W:

W = 36 - L = 36 - 18 = 18.

Therefore, the dimensions of the rectangular pen that will maximize the area are 18 feet by 18 feet.