Minimal surface area for fixed volume

Middle school · ●●●○○ 50 pts

A regular right pentagonal prism has a volume of

278.3105

cubic cm. Let

x

be the side length of the regular pentagonal bases and

y

the length (height) of the prism, in cm. Find the values of

x

and

y

that minimize the total surface area. Use the area of a regular pentagon

\approx 1.720477\,x^2

. Round each value to four decimal places.