A farmer is fencing a rectangular area for cattle and uses a straight portion of a river as one side of the rectangle, as illustrated in the figure. Note that there is no fence along the river. If the farmer has 2400 feet of fence, find the dimensions for the rectangular area that gives the maximum area for the cattle. The width of the rectangular is $\square \mathrm{ft}$. Clear all Skill builder Check answer

Step 1 ：The problem is a classic optimization problem. The farmer has a fixed amount of fencing, and wants to maximize the area of the rectangle. The area of a rectangle is given by the formula width * length.

Step 2 ：Since the farmer is using the river as one side of the rectangle, the amount of fencing used is only for the width and the other length. So, the length of the rectangle is \(2400 - 2*\text{width}\).

Step 3 ：We can substitute this into the area formula to get an equation for the area in terms of width only: area = width * (2400 - 2*width).

Step 4 ：We can find the maximum of this function by taking the derivative, setting it equal to zero, and solving for width.

Step 5 ：The derivative of the area with respect to width is \(2400 - 4*\text{width}\).

Step 6 ：Setting this equal to zero gives a critical point at width = 600 feet.

Step 7 ：Final Answer: The width of the rectangular is \(\boxed{600}\) ft.