Question

An acute triangle has side 8 and 15. How many possible integral lengths are there for the third side?

Answer 1

The third side has to be greater than 7, otherwise a triangle can't be formed.
The third side has to be less than 17, otherwise it will be a right triangle or an obtuse triangle.
So, that leaves us with possible side lengths of 8, 9 10, 11, 12, 13, 14, 15 and 16

We can use the Law of Cosines to determine if Angle B is less than 90 degrees.
cos(B) = (8^2 + 8^2 -15^2) / (2*8*8)
cos(B) = -0.7578125
Angle B = 139.27 Degrees

cos(B) = (9^2 + 8^2 -15^2) / (2*9*8)
cos(B) = -.555555555
cos(B) = 123.75 Degrees

We continue with this until we have side a = 13 where angle B equals 87.796

So, the possible lengths for side "a" are
13, 14, 15 and 16