Question 1203040
<pre>
{{{ 5<= k <= 10 }}} {{{ U }}} {{{ 19<=k<=25 }}}
This amounts to 13 values of k. 

Go to this site:
https://www.triangle-calculator.com/?what=sss&a=15&b=11&c=19&submit=Solve

Plug in 11,15 as two sides and then 'experiment' with other values for the 3rd side.   You will find integer values 5-10 and 19-25 all form obtuse triangles.  The site will draw the triangles so you can see where the obtuse angle is.  
Best wishes. 
.. .. ..
Another approach is to pick a k such that you have a valid triangle.  This means the triangle inequalities must each hold for the value of k.  For this problem:  11+15>k, k+11>15, and k+15>11.   Then, for each triple (11,15,k) pick the longest side and compute the square of each side-length (121,225,k^2).  Call the longest side 'c'.  If  c^2 is greater than the sum of the other two squared values, you have an obtuse triangle.   
Example:
 Say k=19  ... the triangle inequalities hold, check. 
 The values of the squares of each side-lengths are:  (121,225,361)
    Since 361 > 225+121, the triangle is obtuse.
Another example:
 Say k=12...  the triangle inequalities hold, check.
 Squared side-lengths are (121,225,144)
      However, since 225 < 121+144, we do NOT have an obtuse triangle.