The correct answer is "She had made up her mind; she was going to travel the world one way or another."
Explanation:
The most common use of the semicolon (;) is to separate two independent clauses or sentences that can stand alone and express a complete thought. Besides this, semicolons are used to separate long elements of the same type in lists or before some conjunctions such as however, also, therefore, moreover, etc. In the case of the text presented, the only sentence that includes a correct use of semicolon is "She had made up her mind; she was going to travel the world one way or another" because in this the semicolon is used between the two independent clauses "She had made up her mind" and "she was going to travel the world one way or another", considering each of this contains a subject, verb, and complete idea.