21 - 24 Solved problems in maxima and minima

Problem 21
Find the rectangle of maximum perimeter inscribed in a given circle.
 

Problem 22
If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles.
 

Problem 23
Find the most economical proportions for a covered box of fixed volume whose base is a rectangle with one side three times as long as the other.
 

The Polygon

Polygon is a closed plane figure bounded by straight lines. There are two basic types of polygons, a convex and a concave polygon. Polygon is said to be convex if no side when extended will pass inside the polygon, otherwise it is concave.
 

concave-and-convex-polygons.jpg

 

The Quadrilateral

Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. For triangles, the sum of the interior angles is 180°, for quadrilaterals the sum of the interior angles is always equal to 360°
 

$A + B + C + D = 360^\circ$

 

Classifications of Quadrilaterals
There are two broad classifications of quadrilaterals; simple and complex. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.
 

Simple quadrilaterals are further classified into two: convex and concave. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.
 

convex quadrilateral, concave quadrilateral, and complex quadrilateral

 

The following formulas are applicable only to convex quadrilaterals.