# Volume

• Jun 1st 2010, 06:48 PM
sydewayzlocc
Volume
I am stuck on this problem and don't know how to begin...

A right circular cylindrical container is to be built to hold a volume of 500 cubic centimeters. The cost of material used for the top and bottom is \$.5/cm^2 and \$.3/cm^2 . Use calculus to find the radius and cost of the least expensive container.

ANY HELP IS MUCH APPRECIATED
• Jun 1st 2010, 06:56 PM
ANDS!
A couple of questions to ask yourself - What is the formula for the volume of a circular cone? What is the area of a triangle? What is the area of a circle. Write these down, draw a picture, and see if something doesn't click with how to use those three to answer your question (which as you should know, is an optimization problem, and will involve derivatives).
• Jun 1st 2010, 07:41 PM
sydewayzlocc
Quote:

Originally Posted by ANDS!
A couple of questions to ask yourself - What is the formula for the volume of a circular cone? What is the area of a triangle? What is the area of a circle. Write these down, draw a picture, and see if something doesn't click with how to use those three to answer your question (which as you should know, is an optimization problem, and will involve derivatives).

This is what i have so far:
500=pi*r^2h
h=500/pi*r^2
C=\$.5(area of top +bottom)+\$.3(area of side)
=.5(pi*r^2+pi*r^2)+.3(pi2rh)
=pi*r^2+.6pi*r*h
=pi*r^2+.6r(500/pi*r^2)
=pi*r^2+300r^-1
C'=2pi*r-300r^-2
(0=2pi*r-300r^-2)r^2
r=(300/2pi)^1/3=(150/pi)^1/3=15.9155 ft.

and h=(500/pi*r^2)
=500/pi(150/pi)^2/3
=(500/pi)*((pi^2/3)/(500^1/3)
=(500^2/3)/(pi^1/3)=43.01ft

what do i do next?
• Jun 1st 2010, 08:23 PM
ANDS!
Assuming that is correct, now you would simply optimize your function.
• Jun 1st 2010, 08:45 PM
sydewayzlocc
Quote:

Originally Posted by ANDS!
Assuming that is correct, now you would simply optimize your function.

Im still unsure as to how to do that
• Jun 2nd 2010, 02:58 AM
HallsofIvy
Quote:

Originally Posted by sydewayzlocc
This is what i have so far:
500=pi*r^2h
h=500/pi*r^2
C=\$.5(area of top +bottom)+\$.3(area of side)
=.5(pi*r^2+pi*r^2)+.3(pi2rh)
=pi*r^2+.6pi*r*h
=pi*r^2+.6r(500/pi*r^2)
=pi*r^2+300r^-1
C'=2pi*r-300r^-2
(0=2pi*r-300r^-2)r^2
r=(300/2pi)^1/3=(150/pi)^1/3=15.9155 ft.

and h=(500/pi*r^2)
=500/pi(150/pi)^2/3
=(500/pi)*((pi^2/3)/(500^1/3)
=(500^2/3)/(pi^1/3)=43.01ft

what do i do next?

Okay, as long as you are sure that is a minimum and not a maximum or saddle, finish the problem by calculating the cost using those values.