The radius of a cylinder is 325 cm the circumference is 205

The radius of a cylinder is 3.25 cm, the circumference is 20.5 cm and the height is 10 centimeters. The total volume of the cylinder is 331.83 cm3 and the surface area is 270. 57 cm2. How can you minimize surface area and maintain the same volume? Please show work. (without using gm-am)

Solution

Total surface area of cylinder (S) = 2pi*r^2 + 2pi*r*l

where l = height of cylinder and r = radius

volume = pi*r^2*l = 331.83 cm^3

l = 331.83/pi*r^2

S = 2pi*r^2 + 2pi*r*l

= 2pi*r^2 + 2pi*r(331.83/pi*r^2)

= 2pi*r^2 + 663.66/r

find derivatve of S : S\' = 4pi*r - 663.66/r^2 =0

r = (663.66/12.566)^1/3

Minimum occurs at r = (663.66/12.566)^1/3

So, miniumum surface area and volume remains same with r = (663.66/12.566)^1/3 = 3.75 cm