This is an optimization problem that can be solved using linear programming. Let's break it down:

Problem Summary:

• The company makes two types of biscuits: Jumbo and Regular.
• The oven can bake up to 300 biscuits per day.
• Flour constraints: $400 \, \text{oz}$ of flour available.
• Flour usage:
  • Jumbo biscuit: $2 \, \text{oz}$ per biscuit.
  • Regular biscuit: $1 \, \text{oz}$ per biscuit.
• Profit:
  • Jumbo biscuit: $\$0.12$ per biscuit.
  • Regular biscuit: $\$0.07$ per biscuit.

Let:

• $x$ = Number of Jumbo biscuits.
• $y$ = Number of Regular biscuits.

Objective Function:

Maximize profit $P$: $P = 0.12x + 0.07y$

Constraints:

1. Oven capacity: $x + y \leq 300$
2. Flour usage: $2x + y \leq 400$
3. Non-negativity: $x \geq 0$, $y \geq 0$

Solving the System:

We'll solve these linear inequalities to find the values of $x$ and $y$ that maximize the profit.

1. Graphical Method: The feasible region will be a polygon on the coordinate plane.
2. Corner Points: We find the profit at each corner point and choose the one that gives the maximum profit.

Step 1: Find the intersection points of the constraints

• $x + y = 300$
• $2x + y = 400$

We can solve these equations by substitution or elimination to find the intersection points:

Intersection Point:

• Subtract $x + y = 300$ from $2x + y = 400$: $x = 100$ $y = 200$

Step 2: Evaluate the objective function at the feasible points

• $(0, 300)$
• $(200, 0)$
• $(100, 200)$

Calculate Profit $P$:

1. $P(0, 300) = 0.12(0) + 0.07(300) = \$21.00$
2. $P(200, 0) = 0.12(200) + 0.07(0) = \$24.00$
3. $P(100, 200) = 0.12(100) + 0.07(200) = \$26.00$

Conclusion:

The company should make 100 Jumbo biscuits and 200 Regular biscuits to maximize profit, and the maximum income will be $26.00.

Would you like more details, or do you have any other questions?

Related Questions:

1. What would happen if the flour available was reduced to 350 oz?
2. How would the solution change if the profit per Jumbo biscuit increased to $0.15?
3. Can this problem be solved using the Simplex Method instead of the graphical method?
4. How would the solution change if the oven capacity increased to 350 biscuits?
5. What if the flour required for Jumbo biscuits decreased to 1.5 oz per biscuit?

Tip:

Always check the feasibility of each corner point when solving linear programming problems. It ensures the best possible solution within the constraints.