## 41 - 42 Maxima and Minima Problems Involving Trapezoidal Gutter

**Problem 41**

In Problem 39, if the strip is *L* in. wide, and the width across the top is *T* in. (*T* < *L*), what base width gives the maximum capacity?

**Problem 42**

From a strip of tin 14 inches a trapezoidal gutter is to be made by bending up the sides at an angle of 45°. Find the width of the base for greatest carrying capacity.

## 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**

A 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.

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## Length of one side for maximum area of trapezoid (Solution by Geometry)

**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.

## Trapezoidal Strip of Land from a Triangular Lot

**Problem**

A strip of 640 m^{2} is sold from a triangular field whose sides are 96 m, 72 m, and 80 m. The strip is of uniform width h and has one of its sides parallel to the longest side of the field. Find the width of the strip.

A. 7.1 m

B. 8.1 m

C. 8.7 m

D. 7.7 m

**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 m^{2}.

Part 1: Calculate the area of lot ABC.

A. 62,365 m^{2}

B. 59,319 m^{2}

C. 57,254 m^{2}

D. 76.325 m^{2}

Part 2: Calculate the area of lot ADE.

A. 8,342 m^{2}

B. 14,475 m^{2}

C. 6,569 m^{2}

D. 11,546 m^{2}

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°

**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.

The following formulas are applicable only to convex quadrilaterals.

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