Determine the type of shape and

dimensions of the platinum$-$iridium prototype kilogram such that it has the smallest

surface area for a given volume. The standard kilogram is an alloy of $90\%$ platinum and

$10\%$ iridium. Corrosion would affect the mass through chemical reaction; platinum and

iridium were chosen for the standard$$s composition as they resist corrosion. To further

minimize corrosion, the shape should be chosen to have the least surface area. The less

area exposed to the external environment the less chemical change and hence change in

mass can occur.

The density of the alloy is $.$ You may want to consider the following$\backslash$rho $21\mathrm{.}5610-3$

kg

cm $3$

questions. $($a$)$ Is there any reason that the surface area of the standard could be

important? $($b$)$ What is the appropriate density to use? $($c$)$ What shape $($that is$,$ sphere,

cube, right cylinder, parallelepiped, etc.$)$ has the smallest surface area for a given

volume? $($d$)$ Why was a right$-$circular cylinder chosen?