Problem 19 - Essential Calculus

Question

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Step-by-Step Solution

Answer

The dimensions that require the least amount of fencing are a width of \(\sqrt{90,000}\) meters and a length of 600 meters along the river.

Step 1: Understand the Problem

It is given that the farmer is fencing a rectangular field along a river, meaning one of the lengths of the rectangle (the one along the river) doesn't require fencing. Therefore, the fencing is only required for three sides of the rectangle. Let the sizes be x and y. The area of the rectangular field is 180,000 square meters. Now we’ll find the relation between the variables x and y.

Step 2: Formulate the equations

As we've considered the dimensions of the rectangle to be x and y, with y being the side along the river. The area to be fenced A = x * y is given as 180,000 square meters. Hence, the y in terms of x is \(y = \frac{180,000}{x}\). Now, recall that the fence must minimize the perimeter. However, as per the problem, the fencing will be along three sides (2 sides of length x and one side of length y), so the fence's length is P = 2x + y. Using the earlier expression of y, this becomes \(P = 2x + \frac{180,000}{x}\).

Step 3: Find the derivative

To find the minimum of a function, one common method is to find where its derivative is zero and check the results. We need to take the derivative of the function \(P = 2x + \frac{180,000}{x}\) with respect to x and set it to 0. This becomes: \(2 - \frac{180,000}{x^2} = 0\).

Step 4: Solve the derivative equation

To find the x that satisfies \(2 - \frac{180,000}{x^2} = 0\), we first solve for x. The solution gives x = \(\sqrt{90,000}\), which is the length of the shorter sides for which the fence length will be minimized.

Step 5: Calculate the other side

Now that we have the value for x, we use it in our equation \(y = \frac{180,000}{x}\) to find y, the other side. After replacing x, the y becomes 2 x \(\sqrt{90,000} = 2*300 = 600\) meters.