+0  
 
+2
69
4
avatar+290 

In the diagram, \(AOB\) is a quarter circle of radius 10 and \(PQRO\)is a rectangle of perimeter \(26\). What is the perimeter of the shaded region?

TheMathCoder  Apr 25, 2018
edited by TheMathCoder  Apr 25, 2018
 #1
avatar+87301 
+4

Here's my best effort  ....

 

If the perimeter of the rectangle is 26....then 

 

2(W + L )  = 26

W + L  = 13

Then  L  =  13 - W

 

Call point Q  =   ( W , L )  =  (W , 13 - W)

 

So.....using the equation of a circle with a center O located at  (0,0)  and a radius of 10, we have

 

W^2  + (13 - W)^2  =  100

W^2 + W^2 - 26W + 169  = 100

2W^2 - 26W + 69  =  0

 

Solving this for W  we get that W  =  [ 13 + √31] / 2  = OP

And L  =   [ 13  - √31 ] / 2  = PQ 

 

So  PR  =  (1/2) √ [  ( 13 + √31)^2 + (13 - √13)^2 ] =  (1/2) √[ 382 + 26√31 - 26√13 ]

 

So.... the perimeter  =

 

arc BQA  + AP  + PR  + BR  =

 

5pi + (10 - OP) + PR + (10 - PQ)  =

 

5pi  + (10 -  [ 13 + √31] / 2 )  + (1/2) √[ 382 + 26√31 - 26√13 ]  + (10 -  [ 13  - √31 ] / 2 )  ≈

 

33.1125  units

 

 

cool cool cool

CPhill  Apr 25, 2018
edited by CPhill  Apr 25, 2018
 #3
avatar+867 
+3

Oh wow, well done. 

 

I did not think of this method, writing an equation. 

 

I though we make it a whole circle and manipulate it somehow, but I just couldn't find the area of the rectangle. 

GYanggg  Apr 25, 2018
 #2
avatar
+3

CPhill: I will give it a go. Please check my reasoning! Thanks.

 

The diagonal OQ of the rectangle = radius of the circle = 10

Cos(45) =OP / 10 =~ 7.07 =10/sqrt(2)

OR = 26 / 2 - 7.07=~ 5.93 =13 - 5sqrt(2)

10 - OR = BR =~4.07=5sqrt(2) - 3

AP = 10 - 7.07 =~2.93=10 - 5sqrt(2)

Arc AQB =[10 x 2 x Pi] / 4 =~15.71 =5pi

So, the perimeter of the shaded area =10 + [5sqrt(2) - 3] + [10 - 5sqrt(2)] + 5pi =17 + 5pi

Guest Apr 25, 2018
edited by Guest  Apr 25, 2018
edited by Guest  Apr 25, 2018
edited by Guest  Apr 26, 2018
 #4
avatar+290 
+4

Noice guys! Thanks to CPhill and guest!

TheMathCoder  Apr 26, 2018

5 Online Users

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.