ahh i was just doing this last week.
okay so this problem requires optimization.
the first thing we want to do is find the formula for the volume and surface area of a cylinder.
v = πr
2h
sa = 2πr
2 + 2πrh
now we know the volume must equate to 16π (given) therefore 16π = πr
2h
simplified that becomes:
cross out the pi's:
now plug that into the original area formula:
sa = 2πr
2 + 2πr(16/r
2)
then simplify it to: 2πr(r
2+16/r)
now we must derive! 2πr(2r-16/r
2) (using the chain rule for r
2 and quotient rule for 16/r)
set it equal to zero and divide out 2πr: 0 = 2r-16/r
2
simplify more: so r = 8/r
2
multiply both sides by r
2
so r
3 = 8 therefore r = 2
plug that back into
and find that h = 4
the answer is
r = 2 inches and h = 4 inches