Build a rectangular pen with three parallel partitions (meaning 4 total sections) using 500 feet of fencing. What dimensions will maximize the total area of the pen?

Question

Paisley Johnson · Accepted Answer

The dimensions that will maximize the area the total area of the pen will be #125ft.##by# #50ft.#
Total area with these dimensions: #6,250ft.^2#

Step 1: Set up a picture/diagram to help answer the question and write out the needed equations.

Diagram:

The total length will be #x# and the height will be #y#.
Needed Equations:
Perimeter of this diagram #=500=2x+5y#
Total Area#=A=xy#

Step 2: Solve for #y# using the equation #500=2x+5y#
#5y=500-2x#
#y=100-2/5x#

Step 3: Substitute the equation for #y# into the function for area.
#A=x(100-2/5x)#
#A=-2/5x^2+100x#

Step 4: Find the derivative of the equation for area.
#A'=-4/5x+100#

Step 5: Use the derivative equation in order to find the critical point(s) that maximize the area.
Critical points are when #A'=0# and when #A'# does not exist. It is also good to check the endpoints of an equation in order to check for a maximum or minimum.
Since #A'# always exists, only find where #A'=0# (there will be no endpoints to check since this is a pen).

#0=-4/5x+100#
#-100=-4/5x#
#100=4/5x#
#x=125#

#A'# is positive when #x<125# and #A'# is negative when #x>125#, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.

Step 6: Find the height (#y#) when #x=125#.
#500=2(125)+5y#
#5y=250#
#y=50#
The dimensions that will maximize the area the total area of the pen will be #125ft.##by# #50ft.#

Step 7: Find the total area of the pen.
#A=xy#
#A=(125)(50)#
#A=6,250ft.^2#

Isaiah Allen · Answer

undefined To maximize the total area of the pen, the dimensions of the rectangular pen should be as follows:

Length of the pen (parallel to partitions): $ L = 125 $ feet
Width of each section (perpendicular to partitions): $ W = 125 $ feet
Height of partitions: $ H = \frac{500}{4} = 125 $ feet