Question 266652
lets start with
(i) {{{v = (pi)r^2*h}}}
we know v = 50, so we get
(ii) {{{50= (pi)r^2*h}}}
If we solve for h, we get
(iii) {{{h = 50/(pi*r^2)}}}
now we look at surface area as
(iv) {{{sa = 2(pi)*r*h+2(pi)*r^2}}}
substitute (iii) into (iv) to get
(v) {{{sa = 2*pi*r*(50/(pi*r^2)) + 2*pi*r^2}}}
now everything is in terms of r.
getting a common denominator as pir^2, we get
(vi) {{{sa = (100*pi*r + 2*pi^2*r^4)/(pi*r^2)}}}
now, factoring, we get
(vii) {{{sa = (2*pi*r(50+pi*r^3))/(pi*r^2)}}}
reducing we get
(viii) {{{sa = 2(50+pi*r^3)/r}}}
It turns out that r is minimum at 2.
This means that h = 50/4pi or h ~ 3.97887
This gives us a minimum sa at
sa = 75.1326