a farmer has 360 feet of fencing to make three identical adjacent rectangular pens. What dimensions of each pen will maximize the total enclosed area?
To maximize the total enclosed area, we need to find the dimensions of each pen that will give the largest rectangular area. Let's go step by step:
Step 1: Define the variables
Let's assume the length of each pen is "L" and the width of each pen is "W".
Step 2: Understand the constraints
The farmer has 360 feet of fencing, so we can use the perimeter equation to create an equation for the total fencing used: 3L + 4W = 360.
Step 3: Express one variable in terms of the other
To simplify the problem, we need to express one variable in terms of the other. Since we want three identical adjacent rectangular pens, we'll set the length of each pen equal to the width (L = W).
Step 4: Substitute and simplify
Replacing L with W in the perimeter equation, we get: 3W + 4W = 360.
Simplifying the equation, we have: 7W = 360.
Dividing both sides by 7, we find: W = 360/7.
So, the width of each pen is approximately 51.43 feet.
Step 5: Calculate the length
Since L = W, the length of each pen is also approximately 51.43 feet.
Step 6: Calculate the total enclosed area
The total enclosed area of the three identical adjacent rectangular pens is given by: 3LW.
Substituting the values we found, the total enclosed area is approximately: 3 * (51.43) * (51.43) = 7853.71 square feet.
Therefore, to maximize the total enclosed area, each pen should have dimensions of approximately 51.43 feet by 51.43 feet.
To find the dimensions of each pen that will maximize the total enclosed area, we can begin by breaking down the problem into two steps:
Step 1: Determine the dimensions of a single rectangular pen
Step 2: Calculate the total enclosed area
Step 1: Determine the dimensions of a single rectangular pen
Let's assume the length of one pen is L (in feet) and the width is W (in feet). Since all three pens are identical and adjacent, we can consider them as one larger rectangular pen with length 3L (3 pens) and width W (unchanged).
We know that the total fencing available is 360 feet, and we need to consider all four sides of each pen. So, the equation for the fencing is:
4L + 2W = 360
Step 2: Calculate the total enclosed area
The total enclosed area is the sum of the areas of the three pens. To find the area of a single pen, we use the formula:
Area = Length * Width
Since there are three identical pens, the total enclosed area can be calculated as:
Total area = 3 * Area of a single pen
Now, let's substitute the values and solve the equations:
From Step 1:
4L + 2W = 360
From Step 2:
Total area = 3 * (L * W)
Now we have two equations with two variables. We can solve the first equation to express one variable in terms of the other. Let's isolate L:
4L = 360 - 2W
L = (360 - 2W) / 4
Substitute the expression for L in the equation for the total area:
Total area = 3 * ( (360 - 2W) / 4 * W)
Simplifying further:
Total area = (3/4) * (360 - 2W) * W
To find the dimensions that maximize the total area, we need to find the value of W that maximizes this equation. We can do this by taking the derivative of the equation with respect to W and setting it equal to zero. The resulting W value will be the width at which the total area is maximized.
I will calculate the derivative for you:
d(Total area) / dW = (3/4) * (360 - 4W)
Set the derivative equal to zero and solve for W:
(3/4) * (360 - 4W) = 0
Now, solve for W:
360 - 4W = 0
4W = 360
W = 90
Once we have the value of W, we can substitute it back into either equation to find the corresponding value of L:
L = (360 - 2W) / 4
L = (360 - 2 * 90) / 4
L = 90
Therefore, the dimensions of each pen that will maximize the total enclosed area are a length of 90 feet and a width of 90 feet.
so if each pen has dimensions x and y, we have
3x+4y+3x = 360
6x+4y = 360
3x+2y = 180
y = 90 - 3/2 x
the area is
a = 3xy = 3x(90 - 3/2 x) = 270x - 9/2 x^2
da/dx = 270 - 9x
da/dx = 0 when x = 30
So each pen is 30 by 45
As is always the case in such problems, maximum area is achieved when the total fencing is divided equally among lengths and widths.
360/2 = 180
x = 180/6 = 30
y = 180/4 = 45