SOLUTION: A block of metal is cast into a solid cylinder whose volume is 10cm^3. Show that the cylinder has a minimum surface area when r= 3 root (5 over pi). The ends are included.

Algebra ->  Surface-area -> SOLUTION: A block of metal is cast into a solid cylinder whose volume is 10cm^3. Show that the cylinder has a minimum surface area when r= 3 root (5 over pi). The ends are included.      Log On


   

Question 1137934: A block of metal is cast into a solid cylinder whose volume is 10cm^3. Show that the cylinder has a minimum surface area when r= 3 root (5 over pi). The ends are included.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


V+=+%28pi%29%28r%5E2%29%28h%29+=+10 (1)

S+=+2%28pi%29%28r%5E2%29%2B2%28pi%29%28r%29%28h%29 (2)

Solve equation (1) for h in terms of r and substitute into equation (2), giving you an expression in terms of r only for the surface area.

h+=+10%2F%28%28pi%29%28r%5E2%29%29

dS%2Fdr+=+4%28pi%29%28r%29-20%2Fr%5E2

Set the derivative equal to 0 and solve.

4%28pi%29%28r%29-20%2Fr%5E2+=+0
4%28pi%29%28r%29+=+20%2Fr%5E2
r%5E3+=+20%2F%284%28pi%29%29+=+5%2F%28pi%29
r+=+%285%2F%28pi%29%29%5E%281%2F3%29