how do i find the radius and height of a cylinder if I know the Surface Area?

Question

i have a cylinder with surface area of 400cm and i need to know the dimensions that will give maximum volume with minimum surface area

Answer 1

I did a quick thing in Excel and just used solver to maximize the Volume by changing r & h while keeping SA constant at 400.

r=4.606584627
h=9.213193448
V=614.2118213

Note that h = 2 * r = d. This will be the way to maximize the V/SA ratio for any cylinder.

Answer 2

Simple calculus. But you're confused. You say minimum surface area. But you said the surface area was (constant at) 400. Which is it? constant or variable? muym
V=h π r². A=2 πr² + 2 h π r =400
Solve Area equation for h, substitute it into V
differentiate V wrt r, find where dV=0
_
IF you don't know calculus, graph V vs r
the exact maximum volume will be some simple expression with maybe 2 and/or π as part of the factors, not to mention 400 (or powers or roots thereof)

Answer 3

S = 2pir^2+2pirh = 2pir(r+h) = 400 <-- Eq 1
You state that the surface area is 400 cm^2 which is constant.
So that has to be the minimum surface area.
V = pir^2h <-- Eq2
Solve Eq 1 for h getting h= (200 - pir^2)/(pir)
Substitute this in Eq 2 getting V = pir^2(200-pir^2)/(pir)
V = 200r -pir^3
dV/dr =200 -3pir^2
Set this equal 0 and solve for r getting r = sqrt(200/3pi)
r = 4.607 cm
h= (200 - pi4.607^2)/(pi4.607)= 10.45 cm

Answer 4

r = radius of cylinder

h = height of cylinder

Eq#1: A = 2*pi*r^2 + 2*pi*r*h = 400

Eq#2: V = pi*r^2*h

From Eq#1

h = [400 - 2*pi*r^2]/2*pi*r = 200/(pi*r) - r

Substitute to Eq#2

V = pi*r^2*h = r*[400 - 2*pi*r^2]/2

V = -2*pi*r^3 + 200r

V' = -6*pi*r^2 + 200 = 0

r = sqrt(100/(3*pi)) = 3.26 cm

h = 16.28 cm

Answer 5

i forget