Optimisation of surface area in cylinders

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Matilda25

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Hi,
This is my first post so sorry if I have done anything wrong!
I am working on a maths project and I am figuring out the optimal radius that will give you the lowest surface area of a cylinder. I have done the calculus which reveals that the surface area is at a minimum when height is double the radius. I am now trying to find an equation for the relationship between the amount of wasted surface area as a percentage of the minimum surface area and the ratio between height and radius.
If I were to plot it on a graph, the y axis would be the percentage of excess materials needed as a percentage of the minimum possible surface area, and the x axis would be height divided by radius. Since the surface area is minimized when height=2(radius), I know that when x=2, y=0.
Would anyone be able to point me in the right direction as to how to find an equation for this relationship?
Thanks so much!
 
the sketch shows the wasted area of material (in green) for constructing a cylinder with a height h = 2r from a rectangular sheet of material of dimensions [MATH]h \text{ by } (4r + 2\pi r)[/MATH]
your post mentioned “waste” ... I’m just trying to interpret what was meant from that.
 
skeeter said:
the sketch shows the wasted area of material (in green) for constructing a cylinder with a height h = 2r from a rectangular sheet of material of dimensions [MATH]h \text{ by } (4r + 2\pi r)[/MATH]
your post mentioned “waste” ... I’m just trying to interpret what was meant from that.
Click to expand...
The 'wasted surface area' is the minimum possible surface area subtracted from the actual surface area of a can. Does that help? Sorry for the confusion!
 
@Matilda25
Hopefully you can now complete this, finding an expression for y and plotting the graph you require:

1616925038521.png
 
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