Bob and Betty make bird houses and mailboxes in their craft shop near Gatlinburg. Each bird house requires $3 \mathrm{hr}$ of work from Bob and $1 \mathrm{hr}$ from Betty. Each mailbox requires $4 \mathrm{hr}$ of work from Bob and 2 hr of work from Betty. Bob cannot work more than $48 \mathrm{hr}$ per week and Betty cannot work more than $20 \mathrm{hr}$ per week. If each birdhouse sells for $\$ 12$ and each mailbox sells for $\$ 20$, then how may of each should they make to maximize their revenue?
(Hint: Let $x=$ the number of birdhouses and let $y=$ the number of mailboxes, and write one equality for the amount of time Bob can work and another inequality for the amount of time Betty can work. The objective function for this problem is $C=12 x+20 y$ )
Answer
Final Answer: The number of birdhouses they should make is \(\boxed{8}\) and the number of mailboxes they should make is \(\boxed{6}\).
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 \(\boxed{6}\).
