   Chapter 3.7, Problem 28E

Chapter
Section
Textbook Problem

# Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.

To determine

To find:

Area of the largest trapezoid that can be inscribed in a circle of radius 1 whose base is diameter of the circle

Explanation

1) Concept:

First derivative test:

Suppose c is a critical number of a continuous function f defined on an interval.

(a) If f'(x)>0 for all x<c and f'(x)<0 for all x>c, then f(c) is the absolute maximum value of f.

(b) If f'(x)<0 for all x<c and f'(x)>0 for all x>c, then f(c) is the absolute minimum value of f.

2) Formula:

Area of trapezoid

A=a+b2·h

Where a,b are base and h is height

3) Given:

One side of the rectangle lies on the triangle

4) Calculation:

Area of trapezoid=12h(B+b)

Area of trapezoidA=12y2+2x

A=y(1+x)

Let (x,y) be point on circle r=1

x2+y2=1

y2=1-x2

Area maximized when square of area maximized.

T=A2=y21+x2=(1-x2)(1+x2)

Differentiating above equation,

T'=1-x2

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