Acompany manufactures cylindrical paint cans with open tops with a volume of 26,000 cubic centimeters. what should be the dimensions of the cans in order to use the least amount of metal in their production? (round your answer to two decimal places.)

radius cm
height cm

1) Volume, V = Area of the base * heigth = (πr^2)*h = 26,000 cm^3

h = 26,000 / (πr^2)

2) Dimesions of the cans

A = Lateral side + base

Lateral side = 2πrh
Base = π(r^2)

A =  2πrh + πr^2

A = 2πr[26,000/πr^2] + π(r^2) = 52,000/r + π(r^2)

3) Optimization

A' =0

A' = -52,000/r^2 + 2πr =0

-52,000 + 2πr^3 = 0

r^3 = 26,000/π

r = ∛[26,000/π]

r = 20.22 cm
h = 26,000 / [π(20.22)^2] = 20.24  cm

Sum i=1 to 12 of (3i-1):

2+5+...+35

12 terms; sum of first and last terms is 37.

Sum = (number of terms) times (average of all the terms)

For an arithmetic sequence, the average of all the terms is the average of the first and last terms:

Sum = (number of terms) times (average of first and last terms)