Answer:
r = 2,85 feet
h = 8,56 feet
Step-by-step explanation:
Area of two circular caps (m ^(2)) : 2* pi* r^(2) (where r is the radius of circular cap) Area of wall (m ^(2)) is 2*pi*r*h (where h is the height of drum)
V= pi*r^(2)*h so h=V/pi*r^(2) (1)
Cost in $:
Cost of wall 25 $/m^(2) plus 10% for manufacturing so:
C(w) = 27,5*2*pi*r*h $/square feet
Cost of (both top and bottom cap) = 2*40,97*pi* r^(2) so:
C(caps) = 257,29* r^(2) $/squaer feet
Total cost of cylnder = cost of wall + cost of caps
Total cost f cylinder: F(c)= 55*pi*r*h+257,29* r^(2)
F(c) = 172,7*r*h + 257,29*r^(2) F(c) is F(r)
Since from (1) we have h=218,276/pi*r^(2)
F(r) = (172,7)*r*(69,514)/r^(2) +257,29*r^(2) F(c) = 12005/r + 257,29*r^(2)
Taken the first dervative
F´(r)= -12005*(1)/r^(2) +2*257,29*r F´(r) = -12005*(1)/r^(2) +514,58*r
f F´(r) = 0 -12005*(1)/r^(2) + 514,58*r =0
-120005 + 514,58*^(3) = 0 514,58*^(3) = 12005 r =cubic root (23,32)
r = 2,85 ft
if we replace this value en F(r) we can see F(r) tend to infinite both when r tend to 0 and when r tends to infnite so there is a minimun for the functon
r = 2,85 ft and h = 8,56 ft
