# What is a simplified expression to represent the ratio Surface area divided by Volume for a cylindrical can with the height H and the base radius R?What affects the ratio more, a change in height...

What is a simplified expression to represent the ratio Surface area divided by Volume for a cylindrical can with the height H and the base radius R?

What affects the ratio more, a change in height or in the base radius?

Note- This question is based off of a rational expressions unit.

**Surface Area of a Cylinder = `2pir^2+2pi rh` **

**V= `pi r^2h` **

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### 1 Answer

Find a simplified expression for the ratio of the surface area of a cylinder to its volume.

The volume is given by `V=pi r^2h` where h is the height and the radius of the base is r.

The surface area is `S=2pir^2+2pirh`

The ratio of surface area to volume is:

`S/V=(2pir^2+2pirh)/(pir^2h)`

`=(2pir(r+h))/(pir^2h)`

`=(2(r+h))/(rh)`

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To decide whether changing the base radius or the height affects the ratio more you need to know which is bigger.

If the radius and the height are initially the same, then making the same change in the radius and then the height will result in the same change of the ratio.

For example, suppose r=3 and h=3. The ratio is `S/V=12/9=4/3`

If you change r to 4 the ratio becomes `S/V=14/12=7/6` while changing h to 4 results in `S/V=14/12=7/6` .

If one of the parameters is initially larger than the other, then changing the larger changes the ratio more than changing the smaller.