To solve this problem, we can set it up as a Linear Programming Problem (L.P.P.). We need to determine how many cakes of each type can be made given the constraints of flour and fat. Let's define:
Variables
- x: Number of the first kind of cake
- y: Number of the second kind of cake
Objective Function
We want to maximize the total number of cakes:
Maximize Z = x + y
Constraints
Based on the ingredients needed for each cake, we can establish the following constraints:
- Flour: 300x + 150y ≤ 7500 (since 7.5 kg = 7500 g)
- Fat: 15x + 30y ≤ 600
- Non-negativity: x ≥ 0, y ≥ 0
Graphical Solution
To solve this graphically, follow these steps:
- Plot the constraints on a graph with x on the horizontal axis and y on the vertical axis.
- Convert the inequalities into equations to find the boundary lines:
- From 300x + 150y = 7500, simplify to y = 50 - 2x.
- From 15x + 30y = 600, simplify to y = 20 - 0.5x.
- Identify the feasible region where both constraints are satisfied.
- Determine the corner points of the feasible region.
- Evaluate the objective function at each corner point to find the maximum value.
Finding Corner Points
To find the intersection of the two lines:
Set 50 - 2x = 20 - 0.5x and solve for x:
50 - 2x = 20 - 0.5x
30 = 1.5x → x = 20
Substituting back to find y:
y = 50 - 2(20) = 10
So one corner point is (20, 10).
Evaluating Corner Points
Check the other intersections with the axes:
- (0, 50) from the flour constraint
- (40, 0) from the fat constraint
Conclusion
After evaluating the objective function at each corner point, you will find the maximum number of cakes that can be made. The solution will give you the optimal combination of both types of cakes under the given constraints.
