Final answer:
The most cost-effective dimensions for the two rectangular pens with an area of 25 square feet each are squares that are 5 feet on each side. This minimizes the interior fence to 5 feet and the total cost of fencing to $30.
Step-by-step explanation:
To spend the least amount of money on fencing, the dimensions of each pen should be such that the combined perimeter is minimized. Let the length of each pen be l and the width be w. Since each pen has an area of 25 square feet, we have the equation l * w = 25.
The cost of the fencing along the perimeter of the pens is $1/foot, so the cost of the outside fencing is 2l + 2w. The cost of the fencing separating the pens in the interior is $2/foot, so the cost of the interior fencing is 2l.
To minimize the cost, we need to minimize the combined perimeter. We can rewrite the equation for the combined perimeter as C = 2l + 4w.
Substituting the value of l from the area equation into the combined perimeter equation, we get C = 2 * 25/w + 4w. To minimize C, we can take its derivative with respect to w, set it equal to zero, and solve for w. The resulting value of w can then be substituted back into the area equation to solve for l. The dimensions of each pen that will minimize the cost of fencing are w = 5 ft and l = 5 ft.