The problem describes a farmer planning to enclose a rectangular pasture adjacent to a river. The total area of the pasture must be 320,000 square meters, and no fencing is required along the river. The task is to determine the dimensions of the rectangle that minimize the amount of fencing used.

Let's define:

• $x$ as the length of the side parallel to the river (no fencing needed here),
• $y$ as the length of each side perpendicular to the river.

Step 1: Area Constraint

The area of the rectangle must be 320,000 square meters: $A = x \cdot y = 320,000$

Step 2: Fencing to be Minimized

The total amount of fencing required will be for the two sides of length $y$ and one side of length $x$. Therefore, the total fencing required, $F$, is: $F = 2y + x$

Step 3: Use the Area Constraint to Express $y$ in Terms of $x$

From the area equation: $y = \frac{320,000}{x}$

Step 4: Substitute $y$ into the Fencing Equation

Substitute the expression for $y$ into the fencing equation: $F(x) = 2\left(\frac{320,000}{x}\right) + x = \frac{640,000}{x} + x$

Step 5: Minimize the Fencing Function

To find the value of $x$ that minimizes $F(x)$, we take the derivative of $F(x)$ with respect to $x$ and set it equal to zero: $F'(x) = -\frac{640,000}{x^2} + 1 = 0$ $\frac{640,000}{x^2} = 1$ $x^2 = 640,000$ $x = \sqrt{640,000} = 800 \, \text{m}$

Step 6: Find $y$

Now that we have $x = 800 \, \text{m}$, we can find $y$ using the area constraint: $y = \frac{320,000}{800} = 400 \, \text{m}$

Final Answer:

The dimensions that minimize the amount of fencing are: $x = 800 \, \text{m}, \, y = 400 \, \text{m}$

Let me know if you would like further clarification or additional details on any steps!

Here are 5 related questions to consider:

1. How would the problem change if fencing was required on all sides of the pasture?
2. Can you explain why minimizing fencing is important for cost-efficiency?
3. What would happen if the area requirement changed to 500,000 square meters?
4. How does the solution change if the river is not straight, but curved?
5. What calculus principles are used to find the minimum in this problem?

Tip: In optimization problems like this, always use constraints to reduce the number of variables before applying calculus to minimize or maximize the function.