a company is designing a new can for their sodaAs usual the company will create a cylindrical canThey would like the can to have a volume of 113pi cm^3The cost for the aluminum for the top and bottom is .4 cents/cm^2 and the cost for the sides is 2.42 cents/cm^2Find the dimensions of the can that would minimize the cost.
Let a represent the area of the top of the can, and h represent its heightThen the circumference will be c 2√(pia) and the height will be h 113pi/a The price (p) of the aluminum will be p 2(0.4)a + 2.42ch p 0.8a + 2.422√(pia)113pi/a p 0.8a + 3045.43/√a We can find the value of a that minimizes this by setting the derivative of p to zero p' 0.8 - (1/2)(3045.43/a^(3/2)) 0 a (1522.71/0.8)^(2/3) 153.586 The diameter of the can is d 2√(a/pi) 13.98 cm The height of the can is h 113pi/a 2.31 cm _ If the cost of aluminum for top and bottom is higher than for the side, say 4 cents and 2.42 cents, then a more typical beverage container shape is favoredThat can would be 10.7 cm high and 6.5 cm in diameter.
go for a walk a very long walk
