SOLUTION: A company needs to make a cylindrical can that can hold precisely 0.7 liters of liquid. If the entire can is to be made out of the same material, find the dimensions of the can tha

Question 1001753: A company needs to make a cylindrical can that can hold precisely 0.7 liters of liquid. If the entire can is to be made out of the same material, find the dimensions of the can that will minimize the cost.
Found 2 solutions by ankor@dixie-net.com, KMST:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A company needs to make a cylindrical can that can hold precisely 0.7 liters of liquid.
If the entire can is to be made out of the same material, find the dimensions of the can that will minimize the cost.
:
.7 liters = .7*1000 700 cubic/cm
let r = the radius of the can that has a volume of 700 cm/cm
:
= 700
h =
h =
The surface area
S.A. = +
Factor out 2*pi*r
S.A. =
replace h with 222.817/r^2
S.A. =
:
Graph this equation, radius on the x axis, Surface area on the y axis

:
Radius for minimum surface area about 4.8 cm
Find the height
h = 222.817/4.8^2
h = 9.67 cm
:
Check: find the vol with these values
V =
V = 700 cu/cm
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The volume of the can must be

Let's define the measurements
= radius of the base of the can, in cm
= height of the can, in cm
As functions of those two variables, the Volume (in cubic centimeters) and the total surface Area of the can (in square centimeters) are
and .
Since the volume of the can must be ,
--->--->
Substituting the expression fond for in , we get

THE CALCULUS SOLUTION:
The minimum for occurs for a value of that makes the derivative zero.

--->--->--->---> or
For , and is decreasing.
For , and is increasing.
For , and is minimum.
So, the dimensions of the can that will minimize the cost are
and
---> .
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