The problem describes a farmer who has 2400 feet of fencing to enclose a rectangular field, with one side of the rectangle bordering a river (so no fence is needed on that side). The goal is to find the dimensions of the field that would maximize the enclosed area.

Problem Breakdown:

• Let the length of the field along the river be $L$.
• Let the width (perpendicular to the river) of the field be $W$.
• The total amount of fencing is 2400 feet, and the farmer only needs to fence the two widths and one length (since the other length is bordered by the river). Therefore, the fencing constraint is: $2W + L = 2400 \text{ feet}$
• The area $A$ of the rectangular field is: $A = L \times W$

Maximizing the Area:

To maximize the area, we need to express $L$ in terms of $W$ using the fencing constraint, then substitute into the area formula.

1. From the constraint $2W + L = 2400$, solve for $L$: $L = 2400 - 2W$
2. Substitute this into the area formula: $A = (2400 - 2W) \times W$ $A = 2400W - 2W^2$

Optimization:

To find the value of $W$ that maximizes $A$, we take the derivative of $A$ with respect to $W$ and set it equal to zero: $\frac{dA}{dW} = 2400 - 4W = 0$ Solve for $W$: $2400 = 4W \quad \Rightarrow \quad W = 600 \text{ feet}$

Finding $L$:

Now substitute $W = 600$ into the equation for $L$: $L = 2400 - 2(600) = 1200 \text{ feet}$

Conclusion:

The dimensions of the field that maximize the area are:

• Length $L = 1200$ feet
• Width $W = 600$ feet

Would you like further clarification on any of the steps, or have any additional questions on this problem?

Here are some follow-up questions to explore:

1. How would the area change if the fencing was limited to a different amount?
2. How would the dimensions change if the farmer had to fence the entire perimeter (including the river side)?
3. What is the second derivative of the area function, and what does it tell us about the maximization?
4. Can you think of a real-life scenario where this kind of optimization might be used?
5. How could this problem be adjusted if there were other constraints, like specific distance requirements between the fence and the riverbank?

Tip: Always check that the second derivative of a function is negative when trying to find a maximum, as it ensures the function is concave down.