Answer:
Explanation:
(a) Let's denote x as the number of liters of drink A and y as the number of liters of drink B.
From the given information, we can form the following inequalities:
For orange juice: 2x + 4y ≤ 16 (The total grams of orange juice should not exceed 16gm)
For pineapple juice: 3x + 5y ≤ 21 (The total grams of pineapple juice should not exceed 21gm)
We also have the constraints that x and y should be greater than or equal to zero since we cannot have negative quantities of drinks:
x ≥ 0
y ≥ 0
(b) To graphically represent the solution region, we need to plot these inequalities on a graph. The shaded region will represent the feasible solution space.
Here's a graph showing the shaded feasible region:
```
y
^
| 5x + 4y ≤ 16
| 3x + 5y ≤ 21
|__________________
| | /
| | /
|_________|_____/_____________ x
0 4 8
```
The shaded area represents the feasible region that satisfies all the inequalities.
(c) To find the quantity of each type of drink that maximizes profit, we need to define the objective function. Let's assume P(x, y) represents the profit function:
P(x, y) = 10,000x + 15,000y
Now, we need to find the maximum value of P(x, y) within the feasible region.
To do that, we evaluate the profit function P(x, y) at the corner points (vertices) of the feasible region. The point that gives the highest profit value will be the solution.
By examining the graph, we find the vertices of the feasible region are:
(0, 0)
(0, 3)
(4, 0)
(4, 2)
Evaluating the profit function at these points, we get:
P(0, 0) = 10,000(0) + 15,000(0) = 0
P(0, 3) = 10,000(0) + 15,000(3) = 45,000
P(4, 0) = 10,000(4) + 15,000(0) = 40,000
P(4, 2) = 10,000(4) + 15,000(2) = 70,000
From the above calculations, we can see that the maximum profit of 70,000 is achieved when the caterer makes 4 liters of drink A and 2 liters of drink B.