Two sides of an acute triangle are 8 and 15. how many possible lengths are there for the third side, if it is a positive integer?

In an accute triangle the 3 angles are less than 90°.

The minimum lenght of the third side is if the other two sides form the lowest possible angle (close to 0 degrees), in which case the length approximates to zero, so if the number is an integer, the minimum length of the fhird side is 1.

The maximum length of the third side is if it is close to the hypotenuse of a right triangle with legs 8 and 15 = >

hypotenuse^2 = 15^2 + 8^2 = 289 = > hypotenuse = 17. So, the lenght of the third side must be less than 17.

So, the possible lengths of the third side are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16.