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rum2563
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[SOLVED] Optimization Problem
A tin can is to have a given capacity. Find the ratio of height to diameter if the amount of tin (total surface area) is a minimum.
c=pi(r^2)h
surface area = 2pi(r^2)+2h(pi)r
h= c/(pi(r^2))
surface area = 2pi(r^2)+2h(pi)r
= 2pi(r^2)+2pi(r)(c/(pi(r^2)))
= 2pi(r^2) + c(r^-1)
Now, derivate of surface area:
SA` = 4hr - c/(r^2)
0 = ( 4h(r^3)-c )/(r^2)
4h(r^3)=c
r = [tex]\sqrt[3]{c/(4pi)}[/tex]
After this I don't know what to do. Can someone please guide me? Thanks.
Homework Statement
A tin can is to have a given capacity. Find the ratio of height to diameter if the amount of tin (total surface area) is a minimum.
Homework Equations
c=pi(r^2)h
surface area = 2pi(r^2)+2h(pi)r
The Attempt at a Solution
h= c/(pi(r^2))
surface area = 2pi(r^2)+2h(pi)r
= 2pi(r^2)+2pi(r)(c/(pi(r^2)))
= 2pi(r^2) + c(r^-1)
Now, derivate of surface area:
SA` = 4hr - c/(r^2)
0 = ( 4h(r^3)-c )/(r^2)
4h(r^3)=c
r = [tex]\sqrt[3]{c/(4pi)}[/tex]
After this I don't know what to do. Can someone please guide me? Thanks.