Question 994359
The surface area of a closed cylinder, {{{A = 2pi*r*h + 2pi*r^2}}}
where r is the radius, and h is the height
Express h in terms of the volume, V which is known:
{{{V = pi*r^2*h -> h = V/(pi*r^2)}}}
So we can express the area as:
{{{A = 2pi*r*V/(pi*r^2) + 2pi*r^2 -> A = 2V/r + 2pi*r^2}}}
The surface area is minimized when {{{dA/dr = 0 = -2V/r^2 + 4pi*r}}}
Solving for r gives {{{r = (V/(2pi))^(1/3)}}}
V = 40 gallons * 231 ci/gallon = 9240 ci (cubic inches)
Therefore, {{{r = (9240/(2pi))^(1/3) = 11.372 }}}inches
{{{h = 9240/(pi*11.372^2) = 22.744 }}}inches