The Maximization of a Trapezoid Inscribed in a Semicircle

1
The Maximization of a Trapezoid Inscribed in a
Semicircle

• Andrew Gruen

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Basic Maximization Process

• Find the formula for the desired value, all with
  respect to a single variable
• In our case the formula of the trapezoid, defined
  only by the variable x
• Take the derivative of the formula
• Find the critical numbers of the derivative
• Analyze the critical numbers to determine which
  is the maximum
• Use the critical number determined to be the
  maximum in the original area formula

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Specific Trapezoid Maximization

• Draw the trapezoid in the semi circle such that
  the larger base intersects the axis at the points
  (25,0) and (-25,0), and the upper base intersects
  the circle at (x,y) and (-x,y)

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Specific Trapezoid MaximizationContinued

• To find the value of y, draw a radius to the
  point (x,y), thereby creating a right triangle.

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Specific Trapezoid MaximizationContinued

• Use the Pythagorean theorem to find the y value
• Yv(252-x2 )

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Specific Trapezoid MaximizationContinued

• Find the formula to be maximized with respect to
  the single variable, x
• Area 1/2(h)(Basea Baseb)
• h v(252-x2 )
• Basea2x
• Basea2(25)50
• Thus
• Area(1/2)(2x50) v(252-x2 )

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Specific Trapezoid MaximizationContinued

• Take the first derivative of the area function,
  set equal to zero, find critical numbers
• A(x)(1/2)(2x50) v(252-x2 )
• A(x)v(2500- x2 )-(x2 25x)/v(2500- x2 )0
• Then
• x29.654
• Negative x values are discarded because the
  trapezoid cannot have a negative area

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Specific Trapezoid MaximizationContinued

• 6. Test the critical number to be certain it is a
  maximum value

-
29.654
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Specific Trapezoid MaximizationContinued

• Find the maximum area by placing the maximum x
  value back into the area equation
• A(29.654)2200.216