Question 1043644
<pre><b>
The area of the region between the two circles is given to be 40. 

Since {{{OP/OQ}}}{{{""=""}}}{{{3/5}}}{{{""=""}}}{{{(3k)/(5k)}}}, then

Choose k be such that OP = 3k, OQ = 5k, 

then OS = 3k, SQ = 2k

Radius of smaller circle = 3k
Radius of larger circle = 5k

 {{{drawing(400,400,-5.2,5.2,-5.2,5.2, locate(.5,2,3k),
locate(3,0,S), locate(0,0,O),
circle(0,0,3), circle(0,0,5), circle(0,0,.05),
locate(1.5,3,P), locate(1.4,0,3k), locate(4,0,2k),
line(0,0,5,0),locate(5.1,.2,Q), line(0,0,3cos(62*pi/180),3sin(62*pi/180)))}}}

Area of larger circle = {{{pi*(5k)^2}}} = {{{25pi*k^2}}}
Area of smaller circle = {{{pi*(3k)^2}}} = {{{9pi*k^2}}}

Area of region between circles = {{{25pi*k^2-9pi*k^2}}} = {{{16pi*k^2}}}

And since the area of the region between the two circles is given to be 40, 

{{{16pi*k^2}}}{{{""=""}}}{{{40}}}

{{{k^2}}}{{{""=""}}}{{{40/(16pi)}}}{{{""=""}}}{{{5/(2pi)}}}

Area of larger circle = {{{pi*(5k)^2}}} = {{{25pi*k^2}}} = {{{25pi*expr(5/(2pi))}}}{{{""=""}}}{{{125/2}}}{{{""=""}}}{{{62.5}}}

Edwin</pre></b>