   Chapter 3.P, Problem 18P

Chapter
Section
Textbook Problem

# ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas that the triangle AEE can have.

To determine

The maximum and minimum area of the triangle AEF.

Explanation

1) Concept:

i. The function is minimum when f'x<0  and maximum when f'x>0

ii. Pythagorean theorem hypotenus2=one side2+another side2

iii. Power rule of differentiation ddxxn=nxn-1

iv. Chain rule ddxfxgx=fxddxgx+gxddxf(x)

2) Given:

ABCD is a square piece of paper with sides of length 1m. A quarter circle is drawn from B to D with center A. A piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle.

3) Calculation:

From the description in the given statement, the figure is as below:

Let the coordinate after folding the paper is A'(x,y) and AE= x, |AF| = y, so that then the area of triangle AEF is

A=12xy(1)

Now find the relation between x and y to get the derivative dA/dx, and then find the maximum and minimum areas. For that, we can write area(AEF) in a different way.

Here, the point A' is the end point at which the point A ends after fold of paper.

Consider the point P as an intersection of AA and EF.

Note that AA is perpendicular to EF.

Because of the reflection, |AP|=|PA|.

Since the radius of circle is 1, |AA|=1.

Since |AP|+|PA|=|AA|.

From this, |AP|=1/2

Now area of AEF=12EFAP=12x2+y212=14x2+y2..(2)

From equation (1) and (2),

12xy=14x2+y2

4x2y2=x2+y2

4x2y2-y2=x2

(4x2-1)y2=x2

y2=x24x2-1

y=x24x2-1

y=x4x2-1

Now equation (1) becomes

A=12xx4x2-1

A=12x24x2-1

Differentiate this

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