The formula for volume of cylinder is:
V = π r^2 h
since volume is 100 cubic inches, so:
100 = π r^2 h
Rewriting in terms of h:
h = 100 / π r^2
The least amount of metal means least amount of surface
area needed. The surface area of cylinder is:
A = 2 π r h + 2 π r^2
insert the value of h in terms of r:
A = 2 π r (100 / π r^2) + 2 π r^2
A = 200 r-1 + 2 π r^2
Take the 1st derivate and set to 0, dA/dr = 0:
dA/dr = - 200 r-2 + 4 π r
0 = - 200 r-2 + 4 π r
200 / r^2 = 4 π r
r^3 = 50 / π
r = 2.52 inches
So height h is:
h = 100 / π r^2
h = 100 / π (2.52)^2
h = 5.01 inches
Answers:
r = 2.52 inches
h = 5.01 inches
