SOLUTION: a cylindrical vessel without a lid is to be made of metallic lamina of surface area 462 cm square. find the length of its base radius when its capacity is maximum

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Question 1197132: a cylindrical vessel without a lid is to be made of metallic lamina of surface area 462 cm square. find the length of its base radius when its capacity is maximum
Found 2 solutions by math_helper, ewatrrr:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!


Total area of open cyclinder is:

A(r) = +pi%2Ar%5E2+%2B+2%2Api%2Ar%2AL+   where r is radius and L is length of side



The volume is:

V(r) = +pi%2Ar%5E2%2AL+



We can relate L to r by the given information:

  +pi%2Ar%5E2+=+2%2Api%2Ar%2AL+=+462+ ===>  +L+=+%28462+-+pi%2Ar%5E2%29+%2F+%282%2Api%2Ar%29+



So 

+V%28r%29+=++pi%2Ar%5E2+%2A+%28462-pi%2Ar%5E2%29%2F%282%2Api%2Ar%29+



which reduces to:

+V%28r%29+=+231%2Ar+-+%28pi%2F2%29%2Ar%5E3+



Now the derivative of V with respect to r is:

++dV%2Fdr+=+231+-+%28%283%2Api%29%2F2%29%2Ar%5E2+



Setting dV/dr to 0 and solving for r,  +highlight%28+r+=+7%5C.0+%29+ cm 



Check by plugging in some values near 7.0cm for r, and compute corresponding volume:

r = 6.8  ==>  L = 7.4  ==>  V = 1074.9cm^3

r = 7.0  ==>  L = 7.0  ==>  V = 1077.6cm^3   <<< max capacity

r = 7.2  ==>  L = 6.6  ==>  V = 1076.5cm^3

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
Recommend Graphing V = 231r - (pi/2)r^3 as a check for r = 7cm when Volume is at its Maximum
Obvious max is reached at 7cm (to nearest whole number)
www.padowan.dk/download has an excellent graphing capabilities.
Graphing y = 231x - (pi/2)x^3 (y the volume, x the radius of the Cylinder)