Minimizing and optimization related rate problem

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SUMMARY

The discussion focuses on solving a geometric optimization problem involving a triangle with one side measuring 14 cm and an angle of 40°. The objective is to determine the length of side 'X' that minimizes the length of side 'R'. The sine law is suggested as a potential method for formulating the equation for 'R', although it introduces an unknown angle 'θ' that must be less than 140°. The user seeks guidance on the steps to approach this problem and relevant resources for further understanding.

PREREQUISITES
  • Understanding of basic trigonometry, specifically the sine law.
  • Familiarity with optimization techniques in calculus.
  • Knowledge of triangle properties and angle relationships.
  • Ability to manipulate equations involving trigonometric functions.
NEXT STEPS
  • Study the sine law and its applications in triangle problems.
  • Learn about optimization techniques in calculus, particularly for geometric shapes.
  • Explore resources on solving trigonometric equations with unknown angles.
  • Investigate graphical methods for visualizing optimization problems in geometry.
USEFUL FOR

Students studying geometry and calculus, educators teaching optimization methods, and anyone interested in solving trigonometric optimization problems.

Alphax
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I would like to ask for help solving this problem. I've been at this problem for a few hours without making any progress.

There is a picture of the problem attached in this thread.

So, the question is what is the length of side 'X' used in order to get the shortest possible length for side 'R'. The given value for the triangle are 14 cm on one side and a 40° angle.

what are the steps in approaching the problem?
 

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Maybe someone can point to me a source on the internet to understand the concept a bit more?
 

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