# Finding radius from volume that gives the minimum surface area for cylindrical cans

1. Oct 4, 2008

### superhulk

i need MAJOR HELP!!

this is the problem:

The can do tin can company minimizes costs by construckting cans from the least possible amount of material. this company suppplies many different sizes of cans to packing firms. a designer with the company needs to determine the radius that gives the minimum surface area for cylindrical cans with volumes between 100 cm (cubed) and 155 cm (cubed).

basically i need to know how to find the smallest surface area for the following volumes: 100 300 500 700 900 1100 1300 and 1500 cm(cubed)

2. Oct 5, 2008

### symbolipoint

Re: Finding radius from volume that gives the minimum surface area for cylindrical ca

That seems like a mainly APPLIED type Calculus problem. You first need to derive or look in reference material for the surface area for a circular cylinder. Arrange the equation to be surface area as a function of radius. Find the derivative of this surface area function. You are then interested in minimum surface area; find the radius for each case. (someone check my thinking about all that since a long time has passed since I studied Calculus).

3. Oct 5, 2008

### HallsofIvy

Staff Emeritus
Re: Finding radius from volume that gives the minimum surface area for cylindrical ca

You have two equations: volume of a cylnder of radius r and height h is $\pi r^2h$ whch must be equal to one of the values you give, and surface area is $2\pi r^2+ 2\pi rh$ which is what you want to minimize. You can use the first, volume, equation to solve for, say, h, as a function of r and replace that in the second formula.
It might be a good idea to use "V" for the volume rather than one of the numbers you give so you can just plug in a number after you have solved for r and h.

Do you know how to find the max or min of a function by differentiating?