SOLUTION: If a cone has a volume of 500cm^3. What is it's radius and hight with the smallest surface area

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Question 935948: If a cone has a volume of 500cm^3. What is it's radius and hight with the smallest surface area
Found 2 solutions by Alan3354, ankor@dixie-net.com:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Including the base? or just the lateral area?
PS it's = it is

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
If a cone has a volume of 500cm^3.
What is it's radius and height with the smallest surface area.
:
Find the relationship between the height and radius using the volume
%28pi%2Ar%5E2%2Ah%29%2F3+=+500
multiply both sides by 3
pi%2Ar%5E2%2Ah+=+1500
divide both sides by pi%2Ar%5E2
h+=+1500%2F%28pi%2Ar%5E2%29
:
Surface area formula:
S.A = pi%2Ar%2Asqrt%28r%5E2%2Bh%5E2%29
replace h with 1500%2F%28pi%2Ar%5E2%29
S.A. = pi%2Ar%2Asqrt%28r%5E2+%2B+%281500%2F%28pi%2Ar%5E2%29%29%5E2%29
Graph this equation

minimum surface area occurs when r = 7 cm, about 264 sq cm
Find the height
h = 1500%2F%28pi%2A7%5E2%29
h = 9.7 cm
:
See if that checks out by finding the vol with these value
v = 1%2F3*pi%2A7%5E2%2A9.7
v = 498 cu/cm, is that close enough?