A rancher wants to make a rectangular pen to house her animals. She will divide it up with 3 fences down the middle parallel to one side. If she has 14000 ft of fencing to work with, what are the dimensions of the pen of greatest area?
Width =
Length =
What is the greatest area pen that the rancher can make?
Area =
2x+5y=14000
a = xy = x(14000-2x)/5
da/dx = 2800 - 4x/5
da/dx=0 when x = 3500
so, the pen is 3500x1400
Well, if the rancher wants to divide the pen with 3 fences down the middle, that means she will have 4 equal sections in the pen. Let's call the width of one of these sections x.
So, the width of the pen would be 4x, since there are four equal sections.
Now, the length of the pen would be the remaining length after subtracting the width from the total fencing available. There are two lengths and two widths in a rectangle, so we'll need to subtract those from the total fencing.
We know that the total fencing is 14000 ft, so we can write the equation as:
2(length) + 2(width) = total fencing
2(length) + 2(4x) = 14000
Simplifying the equation gives us:
2(length) + 8x = 14000
2(length) = 14000 - 8x
(length) = 7000 - 4x
Now, to find the greatest area, we need to multiply the length and width of the pen.
Area = length * width
Area = (7000 - 4x) * 4x
Area = 28000x - 16x^2
To find the maximum area, we can take the derivative of the area function and set it equal to zero:
d(Area)/dx = 28000 - 32x = 0
32x = 28000
x = 875
So, the width of the pen would be 4 * 875 = 3500 ft and the length would be 7000 - 4 * 875 = 3500 ft.
Therefore, the dimensions of the pen of greatest area are:
Width = 3500 ft
Length = 3500 ft
And the greatest area that the rancher can make is:
Area = (3500 ft) * (3500 ft) = 12,250,000 sq ft
Hope this helps, and good luck to the rancher with her new pen!
To find the dimensions of the pen of greatest area, we can use a quadratic equation. Let's assume that the width of the pen is x and the length is y.
Step 1: Calculate the total length of the three fences down the middle.
Since the fences are parallel to the width, the total length of the fences down the middle is 3 times the length (3y).
Step 2: Calculate the total length of the two remaining sides.
The remaining two sides are the width (x) and the length (y). These two sides will be used twice, so their total length is 2x + 2y.
Step 3: Write an equation for the total fencing.
The total fencing is the sum of the two previous steps:
2x + 2y + 3y = 14000.
Step 4: Simplify the equation.
2x + 5y = 14000.
Step 5: Solve for x in terms of y.
2x = 14000 - 5y.
x = (14000 - 5y) / 2.
Step 6: Calculate the area of the pen.
The area of a rectangular pen is given by A = length × width:
A = x × y.
Substitute the value of x from Step 5:
A = ((14000 - 5y) / 2) × y.
Step 7: Find the derivative of the area.
To find the maximum area, we need to find the critical points of the area function. To do this, calculate the derivative of the area function with respect to y:
dA/dy = (14000 - 5y)/2 - 5y/2.
Step 8: Set the derivative equal to zero and solve for y.
(14000 - 5y)/2 - 5y/2 = 0.
Multiply both sides by 2 to eliminate fractions:
14000 - 5y - 5y = 0.
Combine like terms:
14000 - 10y = 0.
Subtract 14000 from both sides:
-10y = -14000.
Divide both sides by -10:
y = 1400.
Step 9: Substitute the value of y back into the equation for x.
x = (14000 - 5y) / 2
x = (14000 - 5(1400)) / 2
x = (14000 - 7000) / 2
x = 7000 / 2
x = 3500.
The dimensions of the pen of greatest area are:
Width = 3500 ft,
Length = 1400 ft.
Step 10: Calculate the area of the pen with these dimensions.
Area = Length × Width
Area = 1400 ft × 3500 ft
Area = 4,900,000 ft².
Therefore, the greatest area pen that the rancher can make is 4,900,000 ft².
To find the dimensions and area of the pen of greatest area, we can use calculus concepts. Let's start by assigning some variables:
Let's assume the width of the rectangular pen is x ft.
The length of the rectangular pen will be 2x ft since it has 3 fences dividing it into 4 equal sections.
Now, let's calculate the total amount of fencing required for the pen:
The top and bottom of the pen each require a fence of length x.
The 2 vertical sides of the pen each require a fence of length 2x.
The 3 fences dividing the pen require a total length of 3x.
Therefore, the total amount of fencing required is: x + x + 2x + 2x + 3x = 9x ft.
Since the rancher has 14,000 ft of fencing, we can set up the equation:
9x = 14,000.
Solving this equation will give us the value of x, which represents the width of the pen.
Dividing both sides of the equation by 9, we get:
x = 14,000 / 9 ≈ 1555.56 ft.
Since the length of the pen is twice the width (2x), the length will be:
length = 2 * 1555.56 ≈ 3111.11 ft.
To find the area, we simply multiply the width and the length:
Area = width * length = 1555.56 * 3111.11 ≈ 4,836,090.74 sq ft.
Therefore, the dimensions of the pen of greatest area are:
Width ≈ 1555.56 ft.
Length ≈ 3111.11 ft.
And the greatest possible area is approximately 4,836,090.74 sq ft.