Optimization of Area; Inscribed Rectangles

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SUMMARY

The optimization of the area of inscribed rectangles within a semicircle involves calculating the area using the formula 2xy, where x is the base and y is the height. By substituting the semicircle's equation to eliminate the y-term, differentiation with respect to x reveals that the maximum area occurs at x = sqrt(6). Consequently, the optimal dimensions of the rectangle are 2sqrt(6) for the base and sqrt(6) for the height.

PREREQUISITES
  • Understanding of basic calculus, specifically differentiation
  • Familiarity with the properties of semicircles and their equations
  • Knowledge of geometric area calculations
  • Ability to solve equations involving square roots
NEXT STEPS
  • Study the process of differentiation in calculus
  • Explore the equations of semicircles and their applications
  • Learn about optimization techniques in geometry
  • Investigate the relationship between area and dimensions in various shapes
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Students studying calculus, geometry enthusiasts, and anyone interested in optimization problems related to geometric shapes.

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Homework Statement



http://i.minus.com/jbkHm5oH1LfQ1k.png

Homework Equations



The area of a rectangle is its base times its height.

The Attempt at a Solution



The rectangle is inscribed. Its area is 2xy. I can substitute in the equation of the semicircle to get rid of the y-term in my area equation. I can then differentiate with respect to x and find what zeros the derivative (or causes the derivative not to exist) and test the points I find that are on the domain of x (0 to square root of 12).

I find x = sqrt6 as that point that maximizes area and therefore the dimensions must be 2sqrt6 and sqrt6.

http://i.minus.com/jFWGaWoQsb5FX.jpg
 
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