The problem in the image involves maximizing the area of a rectangular field with a fixed amount of fencing, 320 meters, and one side along a river (which doesn't need fencing). Let's break down the question step-by-step.

Given Information:

• The total length of fencing available is 320 meters.
• One side of the rectangle is along the river, so no fencing is needed on that side.
• The two other sides are of equal length (let's call it $x$), and the opposite side (parallel to the river) is $y$.

Steps to Solve:

Part (a): Find a function for the area $A(x)$

• The total perimeter with fencing is made up of 2 sides of length $x$ and one side of length $y$ (since the side along the river doesn't need fencing). $2x + y = 320$
• Solve for $y$: $y = 320 - 2x$
• The area $A(x)$ of the rectangle is: $A(x) = x \cdot y = x(320 - 2x)$ Therefore, the area function is: $A(x) = 320x - 2x^2$

Part (b): Find the value of $x$ that maximizes the area

To find the value of $x$ that maximizes the area, we take the derivative of $A(x)$ with respect to $x$ and set it equal to zero.

$A'(x) = 320 - 4x$ Set the derivative equal to 0: $320 - 4x = 0 \quad \Rightarrow \quad x = 80$

So, the value of $x$ that maximizes the area is $x = 80$ meters.

Part (c): What is the maximum area?

To find the maximum area, substitute $x = 80$ into the area function: $A(80) = 320(80) - 2(80)^2 = 25600 - 12800 = 12800 \, \text{square meters}$

Final Answer:

• (a) The area function is $A(x) = 320x - 2x^2$.
• (b) The side length $x$ that gives the maximum area is 80 meters.
• (c) The maximum area is 12,800 square meters.

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

Here are 5 related questions to expand on this concept:

1. How would the problem change if there was fencing on all four sides of the rectangle?
2. How do we verify that the critical point $x = 80$ is indeed a maximum?
3. Can you find the dimensions of the rectangle if the total amount of fencing was 400 meters instead of 320 meters?
4. What happens to the area if the river doesn't exist, and all four sides must be fenced?
5. Can you generalize the formula for maximizing the area given any total fencing length?

Tip: For maximization problems, the first derivative test is an efficient way to find local maxima or minima, but always check the second derivative or boundary conditions to confirm if it's truly a maximum.