SOLUTION: A cylindrical can is to have volume 300 cubic centimetres. Find its height and its radius if the total surface area (and hence the total amount of material used) is minimum. [The

Algebra ->  Volume -> SOLUTION: A cylindrical can is to have volume 300 cubic centimetres. Find its height and its radius if the total surface area (and hence the total amount of material used) is minimum. [The      Log On


   

Question 233136: A cylindrical can is to have volume 300 cubic centimetres. Find its height and its
radius if the total surface area (and hence the total amount of material used) is minimum.
[The volume is given by V = r2h, and the surface area is S = 2rh + 2r2, where r is the
radius and h is the height.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A cylindrical can is to have volume 300 cubic centimetres. Find its height and its
radius if the total surface area (and hence the total amount of material used) is minimum.
[The volume is given by V = r2h, and the surface area is S = 2rh + 2r2, where r is the radius and h is the height.
:
Find h in terms of the Volume
pi%2Ar%5E2%2Ah = 300
h = 300%2F%28pi%2Ar%5E2%29
:
Surface area
S = 2%2Api%2Ar%2Ah + 2%2Api%2Ar%5E2
Replace h with 300%2F%28pi%2Ar%5E2%29
S = 2%2Api%2Ar%2A%28300%2F%28pi%2Ar%5E2%29%29 + 2%2Api%2Ar%5E2
we can cancel pi*r
S = 2%28300%2Fr%29 + 2%2Api%2Ar%5E2
S = %28600%2Fr%29 + 2%2Api%2Ar%5E2
:
Plot the equation 2%2Api%2Ax%5E2%2B%28600%2Fx%29:
+graph%28+300%2C+200%2C+-4%2C+8%2C+-100%2C+400%2C+2%2Api%2Ax%5E2%2B%28600%2Fx%29%29+
With the help of a TI83, minimum on this graph: x=3.628, (Min SA ~ 248 sq/cm)
:
hence r = 3.628 cm is the radius
:
Find the height
h = 300%2F%28pi%2A3.628%5E2%29 = 7.255 cm is the height
:
:
Check this by finding the volume with this r and h
V =pi%2A3.628%5E2%2A7.255
V = 300.00