Step 1 :Let \(x\) be the number of birdhouses and \(y\) be the number of mailboxes. The time constraints for Bob and Betty can be represented as \(3x + 4y \leq 48\) and \(x + 2y \leq 20\) respectively.
Step 2 :The objective function to maximize their revenue is \(C = 12x + 20y\).
Step 3 :By using a linear programming solver, we can find the optimal solution to this problem.
Step 4 :The optimal solution is to make 8 birdhouses and 6 mailboxes. This will maximize the revenue, which is $216.
Step 5 :This solution satisfies all the constraints. Bob will work for 48 hours and Betty will work for 20 hours.
Step 6 :The solution is also feasible because the number of birdhouses and mailboxes is non-negative.
Step 7 :Final Answer: The number of birdhouses they should make is \(\boxed{8}\) and the number of mailboxes they should make is \(\boxed{6}\).