Question 1023326
{{{drawing(300,200,-4,4,-2,5,rectangle(-4,-1.6,4,4.4),locate(-4,5,E),locate(-2,1.8,p),locate(1.5,1.7,q),locate(-3.7,-1,y), locate(-3.6,2.5,A), locate(-.1,1.8,x),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}

N(A) = 75 = p+x
N(B) = 35 = x+q
n(E) = 200 = p+x+q+y

</pre>
(i)find the smallest possible value of y where y is (AUB)' 
<pre>
That'll be when there's the most possible in the two circles,
which is when their intersection x is the least, or when 
x=0, p=75, q=35, and y=90 (that's also when p and q are the
most they can be, and x is the least it can be.)

{{{drawing(300,200,-4,4,-2,5,rectangle(-4,-1.6,4,4.4),locate(-4,5,E), locate(-2,1.8,p=75),locate(1.5,1.7,q=35),locate(-3.7,-1,y=90), locate(-3.6,2.5,A), locate(-.4,1.8,x=0),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}
</pre>
(ii)the largest possible value of x where x is AnB. 
<pre>
That'll be when all of B is in x which is when q=0, 
x=35, p=40, and y=125.  (That's also when y is the most
it can be, and when p and q are the least they can be) 

{{{drawing(300,200,-4,4,-2,5,rectangle(-4,-1.6,4,4.4),locate(-4,5,E), locate(-2,1.8,p=40),locate(1.5,1.7,q=0),locate(-3.7,-1,y=125), locate(-3.6,2.5,A), locate(-.5,1.8,x=35),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}
</pre>
(iii) find the value of of q if p= 45
<pre>
That'll be when x=30, q=5, and y=120.

{{{drawing(300,200,-4,4,-2,5,rectangle(-4,-1.6,4,4.4),locate(-4,5,E), locate(-2,1.8,p=45),locate(1.5,1.7,q=5),locate(-3.7,-1,y=120), locate(-3.6,2.5,A), locate(-.5,1.8,x=30),red(circle(-sqrt(2),sqrt(2),2)),red(circle(-sqrt(2),sqrt(2),1.95)),red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),
locate(3.4,2.5,B)
 )}}}

Edwin</pre>