Length of one side for maximum area of trapezoid (solution by Calculus)

Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.
 

Trapezoid with one side tangent to the circle

 

Solution
As described by Alexander Bogomolny of cut-the-knot.org, for maximum area of trapezoid, the point of tangency should be at the midline of AB and DC, thus H is the midpoint of BC.
 

032-largest-trapezoid-by-geometry.gif

 

38 - 40 Solved problems in maxima and minima

Problem 38
A cylindrical glass jar has a plastic top. If the plastic is half as expensive as glass, per unit area, find the most economical proportion of the jar.
 

Problem 39
039-figure-39.jpgA trapezoidal gutter is to be made from a strip of tin by bending up the edges. If the cross-section has the form shown in Fig. 38, what width across the top gives maximum carrying capacity?
 

Problem 40
Solve Ex. 39, if the strip is 11 inches wide and the base is 7 inches wide.
 

02 Trapezoidal lot segregated from triangular land Jhun Vert Fri, 05/01/2020 - 11:20 am

Situation
A triangular lot ABC have side BC = 400 m and angle B = 50°. The lot is to be segregated by a dividing line DE parallel to BC and 150 m long. The area of segment BCDE is 50,977.4 m2.
 

Part 1: Calculate the area of lot ABC.
A. 62,365 m2
B. 59,319 m2
C. 57,254 m2
D. 76.325 m2
 

Part 2: Calculate the area of lot ADE.
A. 8,342 m2
B. 14,475 m2
C. 6,569 m2
D. 11,546 m2
 

Part 3: Calculate the value of angle C
A. 57°
B. 42°
C. 63°
D. 68°
 

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.