A farmer wishes to build a fence for 6 adjacent rectangular pens. If there is 600 feet of fencing available, what are the dimensions of each pen that maximizes total pen area?

The image looks like this:
box box box
box box box

also:

If the interior fencing is $3.00 per foot and the perimeter fence is $5.00 per foot, what are the pen dimensions that minimize cost? Assume that each pen encloses 200 square feet of area.

I get 8x+9y=600
ultimately I get x=150
Does anyone get anything different?

To find the dimensions of each pen that maximize total pen area, we can proceed as follows:

Let's assume the length of each pen is x and the width is y.

Since there are 6 pens adjacent to each other, the total length of the inner fences (between the pens) will be 5 times the width of each pen (5y), and the total length of the outer fences (the perimeter) will be 8 times the length of each pen (8x) plus 9 times the width of each pen (9y).

Therefore, we can set up an equation for the total amount of fencing available:

5y + 8x + 9y = 600

Simplifying this equation, we get:

17y + 8x = 600

To maximize the total pen area, we need to express the area of each pen in terms of x and y and then maximize it.

The area of each pen is given by:

Area = length * width = x * y

Since we have 6 pens, the total area can be expressed as:

Total Area = 6 * (x * y)

Now, we can substitute a value for y in terms of x from the equation 17y + 8x = 600.

To do this, rearrange the equation:

17y = 600 - 8x
y = (600 - 8x) / 17

Now substitute this value into the area equation:

Total Area = 6 * (x * ((600 - 8x) / 17))

To find the dimensions that maximize the total pen area, we need to find the values of x and y that make the total area as large as possible. We can do this by finding the critical points of the function and determining whether they are maximum or minimum.

We can differentiate the total area function with respect to x and set the derivative equal to zero to find the critical points.

Differentiating the function:

d(Total Area) / dx = 6 * (300 - 16x) / 17

Setting the derivative equal to zero:

6 * (300 - 16x) / 17 = 0
300 - 16x = 0
16x = 300
x = 18.75

Now, substitute this value of x back into the equation y = (600 - 8x) / 17 to find the corresponding value of y:

y = (600 - 8 * 18.75) / 17
y = 19.1176

Therefore, the dimensions of each pen that maximize the total pen area are approximately x = 18.75 feet and y = 19.1176 feet.

For the second part of the question, to determine the pen dimensions that minimize cost, we need to consider the cost of the interior fencing and the cost of the perimeter fence.

Assuming each pen encloses an area of 200 square feet, we can calculate the length and width of each pen using the formula:

Area = length * width
200 = x * y

Rearranging this equation, we get:

y = 200 / x

Now, let's consider the cost of the interior fencing and the perimeter fence.

The cost of the interior fencing is $3.00 per foot, and since there are four interior fences (between the adjacent pens), the total cost of the interior fencing can be expressed as:

Cost of interior fencing = 4 * (x + y) * 3

The cost of the perimeter fence is $5.00 per foot, and since there are six perimeter fences (the outer sides of the pens), the total cost of the perimeter fence can be expressed as:

Cost of perimeter fence = 6 * (2x + 2y) * 5

To minimize the cost, we need to minimize the total cost, which is the sum of the cost of the interior fencing and the cost of the perimeter fence. Therefore, the total cost function becomes:

Total Cost = 4 * (x + y) * 3 + 6 * (2x + 2y) * 5

Simplifying this equation, we get:

Total Cost = 12x + 12y + 60x + 60y
Total Cost = 72x + 72y

Now, substitute the value of y from y = 200 / x into the total cost function:

Total Cost = 72x + 72(200 / x)

To minimize the cost, find the critical points of this function by differentiating it with respect to x:

d(Total Cost) / dx = 72 - 72 * (200 / x^2)

Setting the derivative equal to zero:

72 - 72 * (200 / x^2) = 0
1 - (200 / x^2) = 0
x^2 - 200 = 0
x^2 = 200
x = sqrt(200)
x = 14.1421

Now, substitute this value of x back into the equation y = 200 / x to find the corresponding value of y:

y = 200 / 14.1421
y = 14.1421

Therefore, the dimensions of each pen that minimize cost, given that each pen encloses an area of 200 square feet, are approximately x = 14.1421 feet and y = 14.1421 feet.

To find the dimensions of each pen that maximizes the total pen area with 600 feet of fencing, we need to optimize the area function with respect to the dimensions of the individual pens.

Let's assume the length of each pen as x and the width as y.

1. Determine the total perimeter of the six pens:
Since there are six adjacent rectangular pens, the total length of each pen side will be the same. So, the total perimeter can be calculated as 2x (for the top and bottom sides) + 5y (for the four vertical sides).

2. Set up the equation for the total perimeter:
The equation will be: 2x + 5y = 600.

3. Express one variable in terms of another:
To make it easier, let's express x in terms of y:
2x = 600 - 5y
x = (600 - 5y)/2

4. Calculate the area of each pen:
Since each pen is rectangular, the area can be calculated as A = length * width. Therefore, the area of one pen will be x * y.

5. Express the area function in terms of one variable:
Substitute the expression for x from step 3 into the area equation:
A = ((600 - 5y)/2) * y

6. Optimize the area function:
To maximize the area, we need to find the maximum value of the area function. We can use calculus to find the maximum. Taking the derivative of the area function with respect to y and setting it equal to zero will give us the critical point(s).

dA/dy = (600 - 5y)/2 - (5/2)y
0 = (600 - 5y)/2 - (5/2)y
0 = 600 - 5y - 5y
0 = 600 - 10y
10y = 600
y = 600/10
y = 60

7. Find the corresponding x value:
Substitute y = 60 into the expression for x from step 3:
x = (600 - 5(60))/2
x = (600 - 300)/2
x = 300/2
x = 150

So, the dimensions of each pen that maximize the total pen area with 600 feet of fencing are 150 feet in length and 60 feet in width.

Regarding the second question about minimizing the cost, we would also need additional information about the cost per foot for the interior fencing and the perimeter fencing. Could you please provide the cost per foot for both types of fencing?