Question

A caterer can make two types of drinks, A and B. A litre 2gm if orange juice and 3gm of pineapple juice. A litre of type B contains 4gm of orange juice and 5gm of pineapple juice. There are not more than 16gm of orange juice and 21gm of pineapple juice. The caterer can make a profit of 10k on 1gm if A and 15k of B. Assuming that the caterer makes x litres of A and y litres of B: (a) Write all the inequalities connecting x and y. (b) Show by shading, the regions satisfying the inequalities in (a). (c) Find the quantity of each type of drink type the caterer must make if she is to maximize profit.​

Answer 1

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.