The Volume of a Cylindrical Tin can...?

Question

The volume of a cylindrical tin can with a top and a bottom is to be 16pi cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height in inches of the can? The answer is 4 i need to know the steps and how to do the problem please.

Answer 1

Let r = radius of the base and h = height of the cylinder



V = pi r^2 h = 16pi

r^2 h = 16 ------------------------------- (1)



S(r,h) = 2pir^2 + 2pi rh ,since from (1) h = 16/r^2 [ top + bottom + curved surface = S ]

S (r)= 2pi r^2 + 2pi r(16/r^2)

S(r) = 2pi r^2 + 2pi (16/r)



S(r) = 2pi r^2 + (32pi /r)



S'(r) = 4pi r - (32pi /r^2)

= 4pi ( r - (8/r^2)) = 0 => r - (8/r^2) = 0

=> r = 8/r^2

= > r^3 = 8

=> r = 2



Since S'(r) = 4pi r - (32pi /r^2) , S" ( r) = 4pi +(64pi/ r^3 )



S"(2) = 4pi +(64pi/ 2^3 ) = 4pi + (64pi/8) =4pi + 8pi = 12pi > 0

Hence , r = 2 is the radius which would minimize the use of tin sheet.

r^2 h = 16

(2^2)h =16

4h =16

h =16/4

h =4

Answer 2

Cylindrical Tin

Answer 3

OVER 9000!!!!