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.
:
Find h in terms of the Volume
{{{pi*r^2*h}}} = 300
h = {{{300/(pi*r^2)}}}
:
Surface area
S = {{{2*pi*r*h}}} + {{{2*pi*r^2}}}
Replace h with {{{300/(pi*r^2)}}}
S = {{{2*pi*r*(300/(pi*r^2))}}} + {{{2*pi*r^2}}}
we can cancel pi*r
S = {{{2(300/r)}}} + {{{2*pi*r^2}}}
S = {{{(600/r)}}} + {{{2*pi*r^2}}}
:
Plot the equation {{{2*pi*x^2+(600/x)}}}:
{{{ graph( 300, 200, -4, 8, -100, 400, 2*pi*x^2+(600/x)) }}}
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/(pi*3.628^2)}}} = 7.255 cm is the height
:
:
Check this by finding the volume with this r and h
V ={{{pi*3.628^2*7.255}}}
V = 300.00