Answer:
20L + 15L + 15(2W) = Cost = C
LW = 3000
W=3000/L
35L + 30(3,000/L) = C
C(L) = 35L + 90,000L^-1
take the derivative of C(L) and set equal to zero, solve for L
C' = 35 - 90000L^-2 = 0
35=90,000/L^2
L^2 = 90,000/35
L= 300/sqr35 = 300sqr35/35= about 50.71 feet of fencing one 1 side and of shrubs on the opposite side
W =3000/300/sqr35 = 10sqr35=59.16 feet of shrubs on 2 sides
W=59.2
L=50.7
WL = square feet
but rounding to one decimal gives WL=3001 square feet
at cost of 30(59.2) + 15(50.7) + 20(50.7)
= 1776 + 1774.5 = $3550.50
but 30(60)+35(50)=1800+1750= $3550.00, 50 cents less
and 60x50=3000 square feet
30(59.16)+35(50.71) = 1774.8 + 1774.85 = $3549.65 the minimum cost with 59.16 by 50.71 feet
rounding errors make a few cents difference
more exact dimensions are 59.16079783 by 50.70925528 feet for minimum cost
for calculus on the absolute minimum, take the 2nd derivative
or look at the end points
60x50 feet is virtually the cost minimizing dimensions.
look at some other simple numbers on either side and you'll find higher cost. It's
a local minimum, but the endpoints show it's also an absolute minimum
Step-by-step explanation:
