A farmer with 700 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

What is the largest possible total area of the four pens?

Let x=shorter side of the big rectangle.
and y=long side of the big rectangle

Fence required = 5x+2y=700 => y=350-5x/2

Total area of pens
A=xy
=x(350-5x/2)

To get the maximum area, we equate dA/dx=0

dA/dx = 350-5x = 0, or

x=70
y=350-5x/2 = 175

Maximum total area=xy=70*175=12250'

Can someone draw the picture?

Answer is 12250

Well, it seems our farmer wants to maximize the area while keeping the fencing intact. Let's put on our thinking hats!

Since there are four pens, we can divide the rectangular area into four equal smaller rectangles, each with the same length. Let's call the length of the smaller rectangle "x".

Now, the length of the entire rectangular area would be "4x" because there are four of these smaller rectangles. The width of the rectangular area will be "y".

To enclose the entire area, we would need three sides of fencing with a length of "4x" each, and one side with a length of "y".

According to the problem, the farmer has 700 feet of fencing. So, we can write an equation: 3(4x) + y = 700.

Simplifying the equation gives us: 12x + y = 700.

Now, we need to express the area in terms of "x" and "y". The area would be "4xy" since we have four small rectangles.

To maximize the area, we need to find the maximum value of "4xy". To simplify things, we can express "y" in terms of "x" from the equation we obtained earlier: y = 700 - 12x.

Now, we substitute this value of "y" in the area equation: Area = 4x(700 - 12x).

Are you ready for a little bit of calculus, my friend? We need to find the derivative of the area equation to find the maximum value. And voila! The maximum area occurs when "x" is equal to 87.5 feet.

So, the maximum area of the four pens is 4 * 87.5 * (700 - 12 * 87.5) square feet.

Now, I would do the math for you, but being a clown, my hands are quite clumsy. I'll leave the number crunching to you!

Good luck, my friend, and remember, even if the answer doesn't make you laugh, the journey of solving it should keep a smile on your face!

To determine the largest possible total area of the four pens, we need to find the dimensions of the rectangle that maximizes the area.

Let's assume the length of the rectangle is L and the width is W. The perimeter of the rectangle is the sum of all the sides, which is given as 700 ft. Therefore, we can write the equation:

2L + 3W = 700

Since we want to express one side of the rectangle in terms of the other, we can rearrange the equation:

2L = 700 - 3W
L = (700 - 3W) / 2

Now, we can calculate the area of the rectangle using the formula:

Area = Length × Width
Area = (700 - 3W) / 2 × W
Area = (700W - 3W²) / 2

To maximize the area, we need to find the derivative of the area equation with respect to W and set it equal to zero:

d(Area) / dW = 0

By differentiating the equation, we get:

(700 - 6W) / 2 = 0
700 - 6W = 0
6W = 700
W = 700 / 6
W = 116.67 ft

Plugging this value back into the equation for L:

L = (700 - 3(116.67)) / 2
L = (700 - 350.01) / 2
L = 349.99 / 2
L = 175 ft

The width of the rectangle that maximizes the area is approximately 116.67 ft, and the length is 175 ft.

Now, let's find the area of the four pens:

Area of one pen = Length × Width
Area of one pen = 116.67 ft × 175 ft
Area of one pen ≈ 20,417.5 ft²

Since we have four pens, the total area of the four pens would be:

Total Area = 4 × Area of one pen
Total Area ≈ 4 × 20,417.5 ft²
Total Area ≈ 81,670 ft²

Therefore, the largest possible total area of the four pens is approximately 81,670 square feet.

so, is there a question somewhere in there?