53 - 55 Solved Problems in Maxima and Minima

Problem 53
Cut the largest possible rectangle from a circular quadrant, as shown in Fig. 40.

Problem 54
A cylindrical tin boiler, open at the top, has a copper bottom. If sheet copper is m times as expensive as tin, per unit area, find the most economical proportions.

Problem 55
Solve Problem 54 above if the boiler is to have a tin cover. Deduce the answer directly from the solution of Problem 54.

32 - 34 Maxima and minima problems of a rectangle inscribed in a triangle Jhun Vert Tue, 05/05/2020 - 12:18 am

Problem 32
Find the dimension of the largest rectangular building that can be placed on a right-triangular lot, facing one of the perpendicular sides.

Problem 33
A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle.

Problem 34
Solve Problem 33 if the lengths of the perpendicular sides are a, b.

21 - 24 Solved problems in maxima and minima Jhun Vert Mon, 05/04/2020 - 10:37 pm

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 Quadrilateral Jhun Vert Sun, 04/26/2020 - 05:33 pm

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.