`y = 9 - |x| , y=0` Find b such that the line y = b divides the region bounded by the graphs of the equations into two regions of equal area.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Given ,

`y = 9-|x| , y = 0`

first let us find the total area of the bounded by the curves.

so we shall proceed as follows

as given ,

`y = 9-|x| , y = 0`

=>` 9-|x|=0`

=> `|x| -9 =0`

=>` |x|=9`

so `x=+-9`

 

the the area of the region is = `int _-9 ^9 (9-|x| -0) dx`

=>`int _-9 ^0 (9+x -0) dx`+`int _0 ^9 (9-x -0) dx`

=>`[9x+x^2 /2]_-9 ^0 + [9x-x^2/2]_0 ^9`

=>`[0]-[-81+81/2] +[81-81/2]-[0]`

=>`81/2 +81/2 =81`

So now we have  to find the horizonal line that splits the region into two regions with area 81/2

as when the line y=b intersects the curve `y=9-|x|` then the area bounded is 81,so

let us solve this as follows

first we shall find the intersecting points

as ,

`9-|x|=b`

`|x|= 9-b`

`x=+-(9-b)`

so the area bound by these curves `y=b` and `y=9-|x| ` is as follows

A= `int _-(9-b) ^(9-b) (9-|x|-b)dx = 81/2`

=>`int _-(9-b) ^0 (9+x-b)dx +int _0 ^(9-b) (9-x-b)dx =81/2`

=>`[9x+x^2/2-bx]_-(9-b) ^0 +[9x-x^2/2-bx]_0 ^(9-b) = 81/2`

=>`[0]-[9(-(9-b))+(-(9-b))^2 /2-b(-(9-b))]+`

`[9((9-b))-((9-b))^2/2-b((9-b))]-[0]=81/2`

=>` [9((9-b))-((9-b))^2/2-b((9-b))]`

`-[9(-(9-b))+(-(9-b))^2/2 -b(-(9-b))]=81/2`

let `t= 9-b`

so

=> `[9(t)-(t)^2/2 -b(t)] -[9(-t)+(-t)^2/2 -b(-t)]=81/2`

=>`[9t-t^2/2 -bt]+[9t-t^2/2 -bt]=81/2`

=>`18t-t^2-2bt =81/2`

but we know half the Area of the region between `y=9-|x|,y=0` curves =`81/2`

so now ,

`18t-t^2-2bt =81/2`

`18t-2bt-t^2 = 81/2`

=>`t(18-2b)-t^2=81/2`

=> `t^2=t(18-2b)-81/2`

=>`t^2 -t(18-2b)+81/2=0`

this is like  quadratic equation

`ax^2+bx+c=0`

so `t = (-b+-sqrt(b^2-4ac))/2a`

  `=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))/2`

but

`t=9-b`

so,

`9-b=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))/2`

=> `18-2b=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))`

=>`+-sqrt((-(18-2b))^2-4*(81/2))=0`

=>`sqrt((-(18-2b))^2-4*(81/2))=0`

=>`(-(18-2b))^2-4*(81/2)=0`

 

=>`(-(18-2b))^2=4*(81/2)`

=>`(-(18-2b))^2=2*(81)`

=>`(-(18-2b))= +- sqrt(2) *9`

=> `-18+2b=+-9sqrt(2)`

=>`2b=+-9sqrt(2)+18`

=>`b=(18+-9sqrt(2))/2`

so `b= (18+-9sqrt(2))/2`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial