SOLUTION: On a math problem I was given, the question gives you the volume of a cylinder which is 64, and then asks you to find the dimensions that would give you the minimum surface area. I

Algebra ->  Volume -> SOLUTION: On a math problem I was given, the question gives you the volume of a cylinder which is 64, and then asks you to find the dimensions that would give you the minimum surface area. I      Log On


   

Question 1077911: On a math problem I was given, the question gives you the volume of a cylinder which is 64, and then asks you to find the dimensions that would give you the minimum surface area. I'm entirely lost on how to find possible dimensions from the information it has given me. The formula to find the volume is π(r^2)(h) but without being given anything else I'm not sure where to begin. Thank you so much for any help you can offer me!
Answer by ikleyn(52906) About Me  (Show Source):
You can put this solution on YOUR website!
.
As you know, the volume of a cylinder is 

V = pi%2Ar%5E2%2Ah, 

where pi = 3.14, r is the radius and h is the height.

In your case the volume is fixed:

pi%2Ar%5E2%2Ah = 64.         (1)


The surface area of a cylinder is 

S = 2pi%2Ar%2Ah+%2B+2pi%2Ar%5E2,    (2)

and they ask you to find minimum of (2) under the restriction (1).


I can rewrite (2) in the form

S(r) = %282pi%2Ar%5E2%2Ah%29%2Fr + 2pi%2Ar%5E2 = 64%2Fr + 2pi%2Ar%5E2.   (1)


The plot below shows the function S(r) = 64%2Fr + 2pi%2Ar%5E2, and you can clearly see that it has the minimum.





Plot y = 64%2Fr+%2B+2%2A3.14%2Ar%5E2


To find the minimum, use Calculus: differentiate the function to get

S'(r) = -64%2Fr%5E2 + 4pi%2Ar = %28-64+%2B+4pi%2Ar%5E3%29%2Fr%5E2

and equate it to zero.


S'(r) = 0   lead you to the equation  4pi%2Ar%5E3 = 64,   which gives 

r = root%283%2C16%2Fpi%29 = root%283%2C16%2F3.14%29 = 1.72 (approximately).


Answer.  r = 1.72 units, h = 64%2F%283.14%2A1.72%5E2%29 units give the minimum of the surface area.