4.+Dessert

=JUST DESSERTS: Five Examples in Five Minutes =

In a sector AOB of radius //r//, draw a small circle of radius //x// with center O. Draw the tangent to the small circle from the vertex B. As //x// is varied, the area S(//x//) of the black part of the figure will also vary. Show that S(//x//) is a maximum when //x// ≅ (293/744)//r//. ||
 * [[image:fan.png width="473" height="279"]] Hidetoshi and Rothman, 2008,Sacred Mathematics: Japanese Temple Geometry, pg 257 ||

How is the area of the shaded region related to the radius of the circle? media type="custom" key="6581337"

Derivation of algebraic relationship:

A W-I-N modification of the problem above. How is the perimeter of the shaded region related to the radius of the circle?

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Derivation of algebraic relationship:


 * [[image:rhombus.png width="476" height="357"]] Hidetoshi and Rothman, 2008,Sacred Mathematics: Japanese Temple Geometry, pg 118 ||

Find //x// in terms of //a// when the area of the rhombus minus the area of the square is maximized || How is the area of the shaded region related to the length of the diagonal?

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Derivation of algebraic relationship:


 * [[image:problem_47.png width="450" height="378"]] Hidetoshi and Rothman, 2008,Sacred Mathematics: Japanese Temple Geometry, pg 120 ||

Maximize //y// as a function of //x// assuming BC = //a// is constant. ||

How is the area of the square related to the length of the vertical (blue) leg?

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 * [[image:problem_37.png width="476" height="302"]] Hidetoshi and Rothman, 2008,Sacred Mathematics: Japanese Temple Geometry, pg 114 ||

Given a rectangle ABCD with AB > BC, circle is inscribed such that it touches three sides of the rectangle, AB, AD, and DC. The diagonal BD intersects the circle at two points P and Q. Find PQ in terms of AB and BC. ||

How is the length of the (red) chord related to the vertical (blue) side of the rectangle?

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Derivation of algebraic relationship: