Superbats Inc. manufactures two different types of wood baseball bats, the Homer-Hitter and the
Big Timber. The Homer-Hitter takes 8 hours to trim and turn on the lathe and 2 hours to finish.
Each Homer-Hitter sold makes a profit of $17. The Big Timber takes 5 hours to trim and turn on
the lathe and 5 hours to finish, and its profit is $29. The total time available for trimming and
lathing is 80 hours. The total available time for finishing is 50 hours.

This problem is about optimizing the production of two types of bats, Homer-Hitter and Big Timber, to maximize profit while considering the available hours for trimming, lathing, and finishing. We can set it up as a **linear programming** problem. Let's define the variables and constraints.

### Variables:
- Let $ x $ be the number of **Homer-Hitter** bats produced.
- Let $ y $ be the number of **Big Timber** bats produced.

### Objective Function:
The goal is to maximize the total profit, which can be expressed as:
\[
\text{Maximize } P = 17x + 29y
\]
Where:
- Profit per Homer-Hitter bat = $17
- Profit per Big Timber bat = $29

### Constraints:
1. **Trimming and lathing constraint**: The total time available for trimming and lathing is 80 hours. The Homer-Hitter takes 8 hours, and the Big Timber takes 5 hours.
   \[
   8x + 5y \leq 80
   \]

2. **Finishing constraint**: The total available time for finishing is 50 hours. The Homer-Hitter takes 2 hours to finish, and the Big Timber takes 5 hours.
   \[
   2x + 5y \leq 50
   \]

3. **Non-negativity constraint**: The number of bats produced cannot be negative.
   \[
   x \geq 0, \quad y \geq 0
   \]

### Linear Programming Model:
Maximize:
\[
P = 17x + 29y
\]

Subject to:
\[
8x + 5y \leq 80
\]
\[
2x + 5y \leq 50
\]
\[
x \geq 0, \quad y \geq 0
\]

This forms a standard linear programming model. The next step would be to solve this system either graphically (by plotting the constraints) or using a method like the Simplex algorithm to find the optimal values for $ x $ and $ y $.

Would you like me to solve this graphically or numerically? Alternatively, I can explain the Simplex method if you're interested.

---

### Related Questions:
1. How can we graphically solve this linear programming problem?
2. What is the Simplex method and how does it apply to this problem?
3. What happens if the available hours for trimming and lathing are increased?
4. How would the solution change if the profit margins for each bat type were different?
5. What if we introduce a third type of bat with different production requirements?

### Tip:
In linear programming problems, always check the feasibility of the constraints before solving to ensure that the problem can have a solution.

Superbats Inc. manufactures two different types of wood baseball bats, the Homer-Hitter and the Big Timber. The Homer-Hitter takes 8 hours to trim and turn on the lathe and 2 hours to finish. Each Homer-Hitter sold makes a profit of $17. The Big Timber takes 5 hours to trim and turn on the lathe and 5 hours to finish, and its profit is $29. The total time available for trimming and lathing is 80 hours. The total available time for finishing is 50 hours.

Learn how to solve a linear programming problem involving the production of two types of baseball bats, Homer-Hitter and Big Timber. The solution involves optimizing profit while accounting for constraints on time for trimming, lathing, and finishing. Perfect for students in Grades 11-12 and undergraduate level.

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