Optimization of a Cylinder's Height and Radius

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SUMMARY

The discussion focuses on optimizing the dimensions of a cylindrical can to maximize its volume while adhering to a surface area constraint of 6 square centimeters. The volume formula is V = πr²h, and the surface area formula, including the top and bottom, is SA = 2πrh + 2πr². Participants emphasize the importance of expressing one variable in terms of the other to simplify the problem, allowing for the application of calculus to find the maximum volume. Two distinct methods for finding the maximum or minimum of a function of one variable are suggested for further exploration.

PREREQUISITES
  • Understanding of calculus, specifically optimization techniques.
  • Familiarity with the formulas for the volume and surface area of a cylinder.
  • Ability to manipulate equations to express one variable in terms of another.
  • Knowledge of the concept of maxima and minima in mathematical functions.
NEXT STEPS
  • Study the method of Lagrange multipliers for constrained optimization.
  • Learn how to apply the first and second derivative tests to find critical points.
  • Explore the relationship between surface area and volume in geometric optimization problems.
  • Investigate real-world applications of cylindrical optimization in packaging design.
USEFUL FOR

Students in mathematics or engineering, particularly those focusing on optimization problems, as well as anyone involved in design and manufacturing of cylindrical containers.

Manni
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A cylindrical can with height h and radius r is to be used to store vegetarian chilli. It
is to be made with 6 square centimetres of tin. Find the height h and radius r which
maximizes the volume of the can.
Hint: The volume of a cylinder is r2h and the surface area of the side walls of a cylinder
is 2rh. The can will also have a top and a bottom, of course — even veggie chilli spoils
if not sealed completely.

I don't know how to solve this question. The mechanics aren't important to me, I'm more concerned of the method. Can somebody explain their methodology for such a question? Thank you!
 
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What are the formulas for volume and surface area? The formula for volume is given. Part of the formula for surface area is given- as the problem says, you need to add the areas of the top and bottom. Both formulas have two variables, the height and radius of the can. Use the information about surface area to solve for one variable and put it into the formula for volume so you have volume depending on one variable.

Do you know how to find max or min of a function of one variable?
(For this particular problem there are two very different methods.)
 
I'm also having trouble with this problem. Given your recommendation, I added the area of the top and bottom to the surface area formula. The top and bottom are just circles so here's what I got:

SA = 2*pi*r*h + 2*pi*r^2

If I solve for h and plug into the volume formula (pi*r^2*h), where do I go from here?
 

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