Question 1089416
I understand that the sector and circle look like the drawing below,
and that the problem asks for one  possible integer value for the length of the arc shown in red.
{{{drawing(300,300,-1.1,1.1,-1.1,1.1,
circle(0,0,0.02),circle(0,0,1),
line(0,0,1,0),line(0,0,-0.5,-0.866),
red(arc(0,0,2.01,2.01,0,120)),
red(arc(0,0,1.99,1.99,0,120)),
locate(1.02,0.05,A),locate(-0.55,-0.87,B),
locate(0.3,0.12,radius),locate(0.4,0,3)
)}}}
The area of a sector or a circle with radius {{{R}}}
and length of arc or circumference {{{L}}} is
{{{area=R*L/2}}} .
(If that reminds you of the formula for area of a triangle, it is not a coincidence).
In this case, we are told that for the sector with arc length {{{AB=L}}}
{{{5<3L/2<10}}} .
Multiplying all three sides of that inequality times {{{2/3}}} , we get
{{{5(2/3)<L<10(2/3)}}}
{{{10/3<L<20/3}}}
{{{3&1/3<L<6&2/3}}} .
So, possible integer arc lengths are {{{L=4}}} or {{{L=5}}} or {{{L=6}}} .