Max Area of Frustrum: Parabola & Line Constraint

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SUMMARY

The discussion centers on finding the maximum area of a frustrum bounded by a parabolic curve defined by the equation y = -x² + 16 and the line y = 0. Participants explore the relationship between the dimensions of the frustrum and the constraints imposed by the parabola. The conversation highlights the importance of understanding geometric properties and ratios in solving such problems, particularly the ratio of radius and height in cone geometry. The proposed area calculation of 4*64/3 is questioned, prompting a deeper exploration of the mathematical principles involved.

PREREQUISITES
  • Understanding of parabolic equations and their properties
  • Knowledge of geometric shapes, specifically frustrums and cones
  • Familiarity with calculus concepts related to area maximization
  • Ability to apply ratios in geometric contexts
NEXT STEPS
  • Study the derivation of area formulas for frustrums and parabolic shapes
  • Learn about optimization techniques in calculus, particularly for geometric problems
  • Explore the relationship between the dimensions of cones and frustrums in detail
  • Investigate practical applications of parabolic shapes in engineering and design
USEFUL FOR

Mathematicians, engineering students, and anyone interested in geometric optimization problems will benefit from this discussion, particularly those working with parabolic shapes and frustrums.

mathwiz123
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Find the maximum area of a frustrum bounded by a paroloid and line y=0. (constraining parabola =-x^2 + 16.)
 
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Is it 4*64/3? (Quick mental calculation).
 
Can you show how you did it? Cause this is something I've thought about after my son showed me his math homework a few years back. Found this forum, and decided to see if you guys knew.
 
Sorry, we don't provide the solutions to homework problems here. If you show your work and explain your reasoning, we can help you when you get stuck. BTW, the "my son showed me this a few years back and I can't stop thinking about it" line is unnecessary -- you aren't going to trick us into doing your homework for you.

- Warren
 
I was actually serious...I understand that you find a lot of kids on the site. I thought of this problem today when trying to build a brace for a parabaloid shaped piece. I thought of a lot of different shapes to put in it. Cylinder, cone, etc. Frustrum came to mind and I thought of the homework problem. I'm sure I could consult any calculus textbook, but you folks seemed to provide interesting responses. If you want my "work" I know the ratio of radius and height of the "small" and "big" cone are equal. But that's all I know.
 

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