# Dimensions of cylinder of a constant surface area but changing volume

• December 28th 2012, 12:20 PM
taursaw62
Dimensions of cylinder of a constant surface area but changing volume
Hi there! I'm new around so please feel free to correct me on forum ettiquete here!

I am modelling a cylinder that changes volume but retains a constant surface area across a range of dimensions. With an increase in volume the cylinder decreases in length and increases in radius and vice versa for a decrease in volume.

So far:
S=2pi*r(r+h) - Surface area of a cylinder (eq1)
V=pi*r2h - Volume of cylinder (eq2)

Rearanging volume for height: h=V/(pi*r^2) (eq3)

Substituting eq3 into eq1 yields:
S=2pi*r(r+V/(pi*r2) = 2pi*r2+2/(V*r) = 2(V*pi*r3+1)/(V*r) (eq4)

After failed attempts to rearrange eq4 and solve for r, I have tried a symbolic solve with matlab but the result is incredibly messy!

Would anyone be able to offer any help as to how I may solve this problem as I expect that the method that I am trying to use is not the best for the job!

• December 28th 2012, 12:43 PM
skeeter
Re: Dimensions of cylinder of a constant surface area but changing volume
$S = 2\pi rh + 2\pi r^2$

$h = \frac{S - 2\pi r^2}{2\pi r}$

$V = \pi r^2 h = \pi r^2 \cdot \frac{S - 2\pi r^2}{2\pi r} = \frac{Sr}{2} - \pi r^3$
• December 29th 2012, 03:24 PM
taursaw62
Re: Dimensions of cylinder of a constant surface area but changing volume
Could you explain how these equations can help me please? I need a equation solving for r in terms of V and S. Again, rearrangement produces a messy result.

Thanks.
• December 29th 2012, 04:50 PM
skeeter
Re: Dimensions of cylinder of a constant surface area but changing volume
well ... you have the cubic equation

$V = \frac{Sr}{2} - \pi r^3$

how are you at using the cubic formula?
• January 4th 2013, 06:38 AM
taursaw62
Re: Dimensions of cylinder of a constant surface area but changing volume
Thats great. Thankyou for your help! Now to implement it!