Question 1203040
.
Please help me with this
The sides of a triangle have lengths 11, 15, and k where k is an integer. 
For how many values of k, is the obtuse triangle?
~~~~~~~~~~~~~~~



        The solution by tutor @math_helper is fine.

        In this post,  I bring another solution.



<pre>
First, from the triangle inequalities, we have these two inequalities for k

    k < 11+15 = 26  and  k > 15-11 = 4.


So, the possible integer values of k are in this interval  5 <= k <= 25.           (1)


Now we will analyze what restrictions come from the condition that the triangle is obtuse.



    If the obtuse angle is between the sides 11 and 15, then this inequality must held

        k >= {{{sqrt(11^2 + 15^2)}}} = 18.6  (approximately),


    which for integer values of k gives  k >= 19. 

    Thus, combining it with (1), in this case, it must be  19 <= k <= 25.          (2)



    If the obtuse angle is attached to one of the sides 11 or 15, then this inequality must held

        k <= {{{sqrt(15^2 - 11^2)}}} = 10.2  (approximately),


    which for integer values of k gives  k <= 10. 

    Thus, combining it with (1), in this case, it must be  5 <= k <= 10.           (3)


From  (2)  and  (3), the final answer is  { 5 <= k <= 10  or  19 <= k <= 25 },
giving 13 possible integer values of "k".
</pre>

Solved.


-------------------


Tutor @greenestamps successfully retold my solution to you in his own words for your better understanding.