+0  
 
0
63
2
avatar+464 

A square sheet of paper has area \( 6 \text{cm}^2\) The front is white and the back is black. When the sheet is folded so that point A rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is A from its original position? Express your answer in simplest radical form.

 

shreyas1  Oct 6, 2018
 #1
avatar+92890 
+1

The sides of the square must be √6 cm  each

 

Let x be the distance between the legs of the black triangle and the side of the square

 

The white area is composed of a square of sides of  x  and two rectangles with dimensions of x and [ √6 - x ]   

 

So.....the white area  =    x^2 + 2x ( √6 - x )

 

And the area of the  black triangle  =  ( √6 - x )^2 / 2  = [ 6 - [2√6]x  + x^2] / 2

 

Since these areas are equal  we have that

 

x^2 + 2x ( √6 - x )  = [ 6  - [2√6]x + x^2] / 2

 

2 [ x^2  + 2x (√6 - x) ] = 6 - [2√6]x + x^2 

 

2x^2 + 4x(√6 - x ]   = 6 - [2√6]x  + x^2

 

x^2 + [4√6] x - 4x^2  = 6 - [2√6]x

 

-3x^2 + [ 6√6]x   = 6

 

3x^2  - (6√6]x  = -6

 

x^2 - [2√6]x = -2     complete the square

 

x^2  - [2√6]x  +  6  =  -2 + 6

 

( x - √6)^2  = 4    take both roots

 

x - √6  = 2        or     x - √6  = -2

 

x = √6 + 2       or     x  =  √6  - 2

(reject)                           (accept)

 

So  ...let the original position of A   = (0, 0)

 

And the new position  of A  =  ( √6  - ( √6  - 2) ,  √6 - ( √6 - 2) )  = ( 2 , 2)

 

And its distance from its original position is

 

√ [ 2^2  +  2^2  ]  =

 

√8  =

 

2√2

 

 

 

cool cool cool

CPhill  Oct 6, 2018
edited by CPhill  Oct 6, 2018
 #2
avatar+464 
0

Thank you CPhill

shreyas1  Oct 8, 2018

11 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.