How to minimize the amount of material used to make a shape of a given volume? A metal can company will make cylindrical shape cans of capacity 300 cubic centimeters. What is the bottom radius of the cans in order to use the least amount of the sheet metal in the production? Accurate to 2 decimal places.
 A: Steps


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*Formulate the question mathematically: out of all the possible cylinders with volume $300\,\text{cm}^3$, we want to find the one with the minimum, i.e. smallest, surface area.

*Note that calculus provides a technique for finding the minimum of a (differentiable) function: for any input $x_0$ where the output $f(x_0)$ is as small as possible, the derivative of the function must be zero at that input; that is, we must have $f'(x_0)=0$.
Since we want to make the surface area take its minimum value, the surface area should be the output of our function.
Since we want to solve for the radius when this occurs, the radius should be the input of our function.

*Look up the formulas for the surface area and volume of a cylinder.

*Using the constraint that the volume of our cylinders must equal $300\,\text{cm}^3$, solve for the height of one of our cylinders in terms of its radius.

*Express the surface area of one of our cylinders as a function of its radius. Thus, if $r$ denotes the radius of one of our cylinders, you should have an expression involving only numbers (e.g. $\pi$) and $r$ on the right side of
$$f(r)=\text{ surface area of a 300 cm}^3\text{ cylinder of radius }r=(\text{some expression})$$

*Take the derivative of this function with respect to $r$, and find the input(s) for which the derivative is 0. Then check by hand at which of these input(s) the function in fact attains its minimum value.
A: Hint:


*

*Write out the expressions for surface area and volume of cylinders. Here they are for reference:


$ A = 2 \pi  r h + 2 \pi r^2 $
$ V = \pi r^2 h $


*

*We already know what the required volume is so we can set $ V = 300 $.

*Can we combine our expressions for $ A $ and $V $ and make progress that way?
ETA: curse my blurred vision!
