a farmer wants to create pens along side his barn for his chicken ducks and geese. he has 600 ft of fencing and wants to create 2 equal spaces for each species. what is the largest area he could have? Also one side is along a house so no fencing is needed there. the image looks like
box box ho
box box us
box box e
3 of length x
1 of length y
A = x y
length of fencing = 3x+y = 600
so y = (600 - 3x)
A = x(600 - 3x) = -3 x^2 + 600 x
dA/dx = -6 x + 600
min or max A when 6 x = 600
or x = 100
then y = 300
A = x y = 30,000 ft^2
or about 3/4 of an acre
To find the largest possible area for the pens, we need to determine the dimensions of each pen. Let's start by dividing the available fencing into equal-length sides.
We have a total of 600 ft of fencing and there are 2 pens. Since we want each species to have an equal space, we can divide the fencing equally between the pens by using 300 ft of fencing for each pen.
Let's assume the length of each pen is 'x'. Since we need two equal spaces for each species, the width of each pen will also be 'x'.
Considering one side of the pens is already along the house, we only need to enclose three sides of each pen.
So, the perimeter of one pen, excluding the side along the house, would be:
Perimeter = 2 * (length + width) - length
= 2 * (x + x) - x
= 2 * (2x) - x
= 3x
Since we have 300 ft of fencing available for each pen, we can set up the equation:
3x = 300
By dividing both sides of the equation by 3, we find that 'x' equals 100 ft.
Now we can calculate the area of one pen by multiplying the length and width:
Area = length * width
= x * x
= 100 ft * 100 ft
= 10,000 sq ft
Since we have two pens, we can double the area to find the total area:
Total Area = 2 * Area
= 2 * 10,000 sq ft
= 20,000 sq ft
Therefore, the largest area that the farmer could have for the pens is 20,000 square feet.