The Regional Farm Bureau (RFB) is preparing a brochure that offers advice about constructing pens for small farm animals, and they want us to be their consultants. They need us to carefully analyze the following situations and provide a detailed report. Then they will use our information to help them write the brochure.
In the example they wish to describe, it is assumed that the farmer has 900 feet of fencing with which to erect a rectangular pen alongside a long, existing fence (so the existing fence forms one side of the pen). Suppose the pen is to be subdivided into four parts in a two-by-two arrangement by including interior fences parallel to the outside boundaries. Then what dimensions make for the largest combined area? What if the farmer subdivides into nine pens in a three-by-three arrangement? What if the farmer subdivides into n2 pens in an n-by-n arrangement?
I don't understand the two by two arrangement.
What you need to do is write an expression for the total amount of fence (outside+inside lengths)=900
it will be in terms of fractions of length, width.
Then you want to maximize area:
Area= Outsidelength*width.
Now in the fence length needed equation, solve for w. Put that in the area equation.
Take the derivative dA/dl, set to zero, and solve for l.
Might as well go for the general nxn case, then we can plug in what we want.
If there are n^2 small pens, with width x and length y, with length parallel to the long fence, then the total fence used is
n(n+1)x + n^2y = 900
y = (900-n(n+1)x)/n^2
A = n^2 xy
= n^2 x (900-n(n+1)x)/n^2
= x(900-n(n+1)x)
= 900x - n(n+1)x^2
dA/dx = 900 - 2n(n+1)x
max at x = 900/[2n(n+1)]
x = 450/[n(n+1)]
y = 450/n^2
To solve this problem, we need to find the dimensions that will result in the largest combined area for the pen. Let's start by analyzing the situation when the pen is subdivided into four parts in a two-by-two arrangement.
1. Subdividing into Four Pens:
We are given that the farmer has 900 feet of fencing. We want to maximize the combined area of the four pens. To solve this, we can use calculus and optimization techniques. Here's the step-by-step process:
Step 1: Setting up the problem:
Let's assume that the long side of the rectangular pen is x feet, and the short side is y feet. Since the pen is divided into four equal parts, each pen will have a width of x/2 and a length of y/2.
Step 2: Expressing the given conditions in terms of the variables:
The total amount of fencing available is 900 feet. Since the pen is rectangular, we have:
Perimeter = 2x + 2(y/2) + 3y = 900
Simplifying, we get: x + 2y = 900
Step 3: Expressing the combined area in terms of the variables:
The combined area of the four pens is the product of the width and the length of each pen. So, the combined area is given by:
Combined Area = 4 * (x/2) * (y/2) = xy/2
Step 4: Expressing one variable in terms of the other using the given condition:
We can rewrite the perimeter equation, x + 2y = 900, to express x in terms of y:
x = 900 - 2y
Step 5: Expressing the combined area in terms of a single variable:
Now, we can substitute the expression for x in terms of y into the combined area equation:
Combined Area = (900 - 2y) * y / 2 = (900y - 2y^2) / 2 = 450y - y^2
Step 6: Maximizing the combined area:
To find the dimensions that result in the largest combined area, we need to find the maximum point of the combined area function. This can be done by taking the derivative of the combined area equation with respect to y, setting it equal to zero, and solving for y.
Taking the derivative, we get:
d(Combined Area) / dy = 450 - 2y
Setting the derivative equal to zero, we have:
450 - 2y = 0
2y = 450
y = 225
Step 7: Finding the corresponding value of x:
Now, we will substitute y = 225 back into the expression x = 900 - 2y to find x:
x = 900 - 2(225)
x = 450
Therefore, the dimensions that result in the largest combined area for the pen when subdivided into four parts in a two-by-two arrangement are x = 450 feet and y = 225 feet.
2. Subdividing into Nine Pens:
The same process can be applied to find the dimensions that result in the largest combined area when the pen is subdivided into nine pens in a three-by-three arrangement. The only difference is that we will change the number of subdivisions and pens.
Step 1: Setting up the problem:
Let's assume that the long side of the rectangular pen is x feet, and the short side is y feet. Since the pen is divided into nine equal parts, each pen will have a width of x/3 and a length of y/3.
Step 2: Expressing the given conditions in terms of the variables:
The total amount of fencing available is 900 feet. Similarly, we have:
Perimeter = 2x + 4(y/3) + 5y = 900
Simplifying, we get: x + (4/3)y + 5y = 900
Step 3: Expressing the combined area in terms of the variables:
The combined area of the nine pens is given by:
Combined Area = 9 * (x/3) * (y/3) = xy/3
Step 4: Expressing one variable in terms of the other using the given condition:
We can rewrite the perimeter equation to express x in terms of y:
x = 900 - (4/3)y - 5y
Step 5: Expressing the combined area in terms of a single variable:
Now, substitute the expression for x in terms of y into the combined area equation:
Combined Area = (900 - (4/3)y - 5y) * y / 3 = (900y - (4/3)y^2 - 5y^2) / 3 = (300y - (4/3)y^2 - 5y^2) / 3
Step 6: Maximizing the combined area:
Take the derivative of the combined area equation with respect to y, set it equal to zero, and solve for y.
Taking the derivative, we get:
d(Combined Area) / dy = 300 - (8/3)y - 10y
Setting the derivative equal to zero, we have:
300 - (8/3)y - 10y = 0
24 - (8/3)y = 10y
(42/3)y = 24
y = 24 * (3/42)
y = 24 / 7
Step 7: Finding the corresponding value of x:
Substitute y = 24/7 back into the expression x = 900 - (4/3)y - 5y to find x:
x = 900 - (4/3)(24/7) - 5(24/7)
x = 900 - 32/7 - 120/7
x = (6300 - 32 - 120) / 7
x = (6300 - 152) / 7
x = 6148 / 7
Therefore, the dimensions that result in the largest combined area for the pen when subdivided into nine pens in a three-by-three arrangement are x = 6148/7 feet and y = 24/7 feet.
3. Subdividing into n^2 Pens in an n-by-n Arrangement:
Following the same approach, we can generalize the solution for any value of n. The variables x and y will represent the dimensions of the rectangular pen, and both sides will be divided into n equal parts.
The perimeter equation will be:
Perimeter = 2x + 2((n - 1)(y/n)) + n(y/n) = 900
Simplifying, we get: x + (n - 1)(y/n) + y = 900
The combined area equation will be:
Combined Area = n^2 * (x/n) * (y/n) = xy/n
From here, you can proceed with steps 4, 5, 6, and 7 to find the dimensions that result in the largest combined area for any value of n.