Let's use the Law of Cosines to solve this problem.
We know that:
AC^2 = AB^2 + BC^2 - 2(AB)(BC)cos(angle B)
Since angle B is obtuse, cos(angle B) is negative. Therefore, we have:
AC^2 = (7A)^2 + (10B)^2 + 2(7A)(10B)cos(angle B)
AC^2 = 49A^2 + 100B^2 + 140ABcos(angle B)
We are asked to find which of the following cannot be the length of AC. Let's check each option:
A) 10
If AC = 10, then AC^2 = 100. We can substitute this into our equation:
100 = 49A^2 + 100B^2 + 140ABcos(angle B)
We know that A, B, and cos(angle B) are all positive, so the left-hand side of the equation is less than the right-hand side. Therefore, 10 can't be the length of AC.
B) 13
If AC = 13, then AC^2 = 169. We can substitute this into our equation:
169 = 49A^2 + 100B^2 + 140ABcos(angle B)
We know that A, B, and cos(angle B) are all positive, so the left-hand side of the equation is greater than the right-hand side. Therefore, 13 can be the length of AC.
C) 15
If AC = 15, then AC^2 = 225. We can substitute this into our equation:
225 = 49A^2 + 100B^2 + 140ABcos(angle B)
We know that A, B, and cos(angle B) are all positive, so the left-hand side of the equation is greater than the right-hand side. Therefore, 15 can be the length of AC.
D) 17
If AC = 17, then AC^2 = 289. We can substitute this into our equation:
289 = 49A^2 + 100B^2 + 140ABcos(angle B)
We know that A, B, and cos(angle B) are all positive, so the left-hand side of the equation is greater than the right-hand side. Therefore, 17 can be the length of AC.
Therefore, the answer is A) 10.