# To prove that when the system reaches equilibrium the value of x = r 4 d ( r + r 2 + 8 d 2 )

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 21P
To determine

## To prove that when the system reaches equilibrium the value of x=r4d(r+r2+8d2)

Expert Solution

x=r4d(r+r2+8d2)

### Explanation of Solution

Given:

The given figure is

Calculation:

Consider the following diagram:

Let

a=|EF|b=|BF|

l=|BF|+|BD|,|FD|=lb

Now

|ED|=|EF|+|FD|=a+lb=r2x2+l(dx)2+a2=r2x2+ld22dx+r2

Let

f(x)=r2x2+ld22dx+r2f'(x)=12(r2x2)12(2x)12(d2+r22dx)12(2d)=xr2x2+dd2+r22dx

Equate f'(x)=0 , then

xr2x2=dd2+r22dx(xd)[2dx2r2(x+d)]=0

But d>r>x , so xd

Solve the equation (xd)[2dx2r2(x+d)]=0 for x.

x=(r2)±(r2)24(2d)(dr2)2(2d)x=r2±r4+8d2r24dx=r4d(r+r2+8d2)

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