Answer:
Radius = 4.25 cm
Height = 6.26 cm
C = $0 05 per can
Step-by-step explanation:
Volume of a typical soft drink in U.S = 355cm^3
The cost of the two circular ends per cm^2 = $0.0006
The cost of aluminum for the side of the can per cm^2 = $0.00026
Total surface area of can(A)
= 2πr^2 + 2πrh
Volume (V) = πr^2h
πr^2h = 355
h = 355/ πr^2
The cost function = C
C = 0.0006(2πr^2) + 0.00026(2πrh)
C = 0.0012πr^2 + 0.00052(πr*355/πr^2)
C = 0.0012πr^2 + 0.1846/r
To minimize cost, differentiate C with respect to r
dC/dr = 2(0.0012πr) - 0.1846/r^2
= 0.0024πr - 0.1846/r^2
To minimize cost, dC/dr = 0
0.0024πr - 0.1846/r^2 = 0
0.0024πr = 0.1846/r^2
(0.0024πr) r^2 = 0.1846
0.0024πr^3 = 0.1846
r^3 = 0.1846/0.0024
r^3 = 76.91667
r = cuberoot if 76.91667
r = 4.25 cm
Recall that h = 355/πr^2
h = 355 / π(4.25)^2
h = 6.26 cm
Recall that
C = 0.0012πr^2 + 0.1846/r
C = 0.00012π(4.25)^2 + 0.1846/4.25
C = 0.0068094 + 0.043435
C = 0.050244
C = $ 0.05 per can (approximate to 2 d.p)
