Final answer:
To provide each of the 512 llamas with 100,000 square feet of grazing area, you would need a minimum of 21,466.26 feet of fencing for a pen arranged along a river, assuming a perfectly square pen which minimizes the perimeter.
Step-by-step explanation:
You are designing a pen for 512 llamas and require that each llama has 100,000 square feet of grazing area. To calculate the minimum amount of fencing needed, we first determine the total area required for the llamas and then find the dimensions of the pen that would minimize the fence length.
First, we calculate the total area needed for the llamas:
- Total area = Number of llamas × Area per llama = 512 × 100,000 square feet = 51,200,000 square feet.
Since the pen will be rectangular and one side will be along a straight river that does not require fencing, the problem becomes minimizing the perimeter of the rectangle that has a given area. To minimize fencing, we want to create a pen that is as square as possible because a square has the smallest perimeter for a given area.
Let x be the length of the pen that is perpendicular to the river, and y be the length along the river. The area A of the rectangular pen is A = x × y. Since we know A is 51,200,000 square feet, y = 51,200,000 / x.
The perimeter P that needs to be fenced (only three sides because one side is the river) is P = 2x + y. Substituting y using the area, we get P = 2x + (51,200,000 / x).
To minimize P with respect to x, we set the derivative dP/dx to zero and solve for x. Without going through the complete calculus here, we can say that the minimum occurs when x equals y. Thus, we find x by taking the square root of the area:
- x = √(51,200,000 square feet) ≈ 7,155.42 feet (rounded to two decimal places).
Since x equals y, we then calculate the minimum perimeter needed for fencing:
- Minimum P = 2x + y = 2 × 7,155.42 + 7,155.42 = 21,466.26 feet (rounded to two decimal places).