What is the maximum possible volume of a cone with a given height and radius?

  • Context: Undergrad 
  • Thread starter Thread starter Defaultjoe
  • Start date Start date
  • Tags Tags
    Cone Maximum Volume
Click For Summary
SUMMARY

The maximum possible volume of a cone with a height defined as h = 6 - r can be calculated using differentiation. The volume formula V = (πr²h)/3 simplifies to V = (π/3)(6r² - r³). By taking the derivative, V' = (π/3)(12r - 3r²), and setting it to zero, the critical points can be found to determine the maximum volume. The valid range for r is between 0 and 6, excluding these endpoints as they yield minimum volumes.

PREREQUISITES
  • Understanding of calculus, specifically differentiation
  • Familiarity with the formula for the volume of a cone
  • Knowledge of critical points and their significance in optimization
  • Basic algebra for manipulating equations
NEXT STEPS
  • Study the principles of differentiation in calculus
  • Explore optimization techniques in calculus
  • Learn about the geometric properties of cones
  • Investigate real-world applications of volume optimization in engineering
USEFUL FOR

Students and professionals in mathematics, engineering, and physics who are interested in optimization problems and volume calculations.

Defaultjoe
Messages
1
Reaction score
0
I'm machining a component as a means of testing one of our companies new machines. The objective is to manufacture a cone of maximum possible volume. The volume of a cone is given by: pir2h/3.

Given that h = 6 - r, how am i to calculate the maximum possible volume by means of integration.

Many thanks, Joe.
 
Physics news on Phys.org
You can't "by means of integration! You can "by means of differentiation". Putting h= 6- r in [itex]\pi r^2h/3[/itex] gives [itex]\pi r^2(6-r)/3= \pi/3(6r^2- r^3)[/itex]. The derivative of that is [itex]\pi/3(12r- 3r^2)= \pi(4r- r^2)[/itex]. The maximum possible volume is given by the (non-zero) r value that makes that 0.

(neither r nor h can be negative which mean r cannot be less than 0 or greater than 6. Technically we should consider the endpoints 0 and 6 but those obviously give minimum volume (a flat disk or a straight line).
 

Similar threads

Undergrad Surface of a cone by integration
  • · Replies 8 ·
Replies
8
Views
3K
MHB Maximum Volume of a Right Circular Cone with Given Slant Height?
  • · Replies 1 ·
Replies
1
Views
3K
MHB What is the Height of a Cone Confined to a Hemisphere with Given Radius?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Calculate new height of truncated cone
  • · Replies 4 ·
Replies
4
Views
5K
MHB What Is the Optimal Size of a Cylinder in a Cone to Maximize Volume?
  • · Replies 14 ·
Replies
14
Views
3K
MHB 3) Calculate the dimensions of the straight circular cone, smaller volume that can be circumscribed
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
4K
MHB *Volume of a cone change of rate of volume with respect to h and r
  • · Replies 8 ·
Replies
8
Views
35K
MHB What are the methods for finding the volume of a solid of revolution?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Attempt at Deriving the Volume of a Cone
  • · Replies 2 ·
Replies
2
Views
4K