A farmer wishes to enclose a rectangular field of area 450 ft using an existing wall as one of the sides. The cost of the fence for the other 3 sides is $3 per foot. How do you find the dimensions that minimize the cost of the fence?

Answer 1

#a=15, b=30#

The cost function is:

#y(a,b)=3(2a+b)#
since farmer is using the wall of length #b#. The area of the field is:
#450=a*b => b=450/a#
#y(a)=3(2a+450/a)=6a+1350/a#
#y'(a)=6-1350/a^2=(6a^2-1350)/a^2#
#y'(a)=0 <=> 6a^2-1350=0 <=> 6a^2=1350#
#a^2=1350/6=225 => a=15# since #a>0# because it's length.
#AAa in (0,15): y'<0# and #y# is decreasing #AAa in (15,oo): y'>0# and #y# is increasing
So, for #a=15# function #y# has a minimum value #y_min=y(15)=180#

The dimensions of the field:

#b=450/a=450/15=30#
#a=15, b=30#
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