Bob is not authorized, but he may legally do so (in the US, you don't need specific authorization to perform a non-forbidden act). In creating the manual, Bob would need to avoid reproducing the questions (which are protected by copyright), and just give the answers. Copyright protection for the textbook includes not just the actual words, but also organizational structure, so Bob would need to avoid copying the structure of the exercises.
As for "structure of the exercises", textbooks often have exercises for each chapter, with a clear logic to the order and content of chapters. Within the chapter exercises, there is often some rationale to the order of presentation of the dozen exercises, like starting for with simplest concepts presented in the text and moving up the ladder. In a given exercise, there my be subparts, where understanding part 1 leads you to understanding part 2, and so on. In a well-structure textbook, maybe 20% of the creative nature of the book is that artful ordering. Imposing an entirely different order on your solutions avoids copying the author's protected expression.