A cylinder shaped can needs to be constructed to hold 300 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost. A cylinder shaped can needs to be constructed to hold 300 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

Step-by-step explanation:

Cost = Area Cylinder body* cost * 0.03 cm^3 + Area of 2 lids*cost 0.06 of cm^3

Volume = 300 cm^3

Volume = Base *h  

Volume = pi * r^2 * h = 300

h = 300/(pi * r^2 )

Area material =  2*pi * r * h * 0.03 + 2 pi r^2 * 0.06

Area material = 0.06*pi * r * 300/(pi* r^2) + 0.12 * pi * r^2

dmaterial/dr = (-1)18 r^-2 + 0.12 pi * 2*r

dmaterial/dr = -18 r^-2 + 0.24 pi * r

The minimum occurs when the right side = 0

18/r^2 = 0.24 *pi r

18 = 0.24*pi * r^3

18/(0.24 * pi) = r^3

23.89 = r^3

cube root (23.89) = cuberoot(r^3)

r = 2.88 cm

h = 300/pi * r^2

h = 300/(3.14 * 2.88^2)

h = 11.52

This isn't much of a check but we can try it.

Volume = 3.14 * 2.88^2 * 11.52

Volume = 302.01 which considering all the rounding is pretty close.

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