Farmer Jones has 210 meters of fence. She wishes to construct a rectangular pen using the river as one side and dividing the pen into six separate rectangular pens as shown. What should the dimensions of the combined pens be to maximize the area enclosed?
7 vertical and one horizontal connecting them
Why did the farmer want to divide the pen into six separate rectangular pens? Because he heard they were having a "dé-cow-rative" party and each pen needed its own "moo-sic" area!
But let's get serious now. To maximize the area enclosed, we need to find the dimensions that will give us the largest possible area. Let's assume the river side is the length of the combined pens and the connecting side is the width.
Since there are seven vertical pens and one horizontal pen, we can say that the length of the combined pens is 7x and the width is x.
So, the total length of the fence will be 7x + 2x = 210 meters (since we have 7 vertical pens and 2 connecting sides).
Simplifying the equation, we have 9x = 210, which gives us x = 23.33 meters (rounded to two decimal places).
So, the width of the combined pens should be approximately 23.33 meters, which means the length of the combined pens (along the river) should be approximately 7 x 23.33 = 163.33 meters.
Therefore, the dimensions of the combined pens to maximize the area enclosed would be approximately 23.33 meters by 163.33 meters.
Now that's what I call "a-moo-zing" pen planning!
To find the dimensions that would maximize the area enclosed by Farmer Jones' combined rectangular pens, we can use the concept of calculus and optimization.
Let's assume the width of each rectangular pen is 'x' meters, and the length of the horizontal connecting pen is 'y' meters. Since there are a total of 7 vertical pens, the total length of these pens would be 7x meters.
Given that Farmer Jones has 210 meters of fence, the sum of the perimeters of all the pens should be equal to 210.
Perimeter of each vertical pen = 2(length + width) = 2(7x + x) = 16x
Perimeter of the horizontal connecting pen = 2(length + width) = 2(y + x)
So, the equation representing the sum of the perimeters would be:
16x + 2(y + x) = 210
Now, let's express the length 'y' in terms of 'x' to be able to write the area equation. As given in the problem, the river acts as one of the sides of the pens. So, the total length of the horizontal connecting pen is equal to the sum of the width of each vertical pen:
y = 7x
The area of the rectangular pens can be calculated as the product of the length and width of each pen. We want to maximize this area. The total area can be found by multiplying the length and width of each individual pen and summating them:
Total Area = 7x * x + x * y = 7x^2 + 7x^2 = 14x^2
To maximize the Total Area, we need to find the value of 'x' that makes the derivative of the Total Area equation equal to zero. Thus, we differentiate the equation:
d(Total Area)/dx = 28x
Setting the derivative equal to zero:
28x = 0
x = 0
Since x = 0 does not satisfy the original constraint, we can conclude that there is no maximum area for the given constraints. Therefore, the problem might have been stated incorrectly, or there could be another factor missing that affects the optimization process.