a. A farmer raises pigs and goats in a pen which can house at most 16 animals. She plans not to have more

than 12 goats. If she spends Gh₵50.00 to raise a pig and Gh₵20.00 to raise a goat and has GH₵500.00
available for this project, find the maximum profit she can make if she makes a profit of GH₵80.00 per goat
and GH₵40.00 per pig.

1 answer

Let's assume the farmer has x pigs and y goats.

From the problem, we know that:
- x + y ≤ 16 (the pen can house at most 16 animals)
- y ≤ 12 (she plans not to have more than 12 goats)
- 50x + 20y ≤ 500 (she has GH₵500.00 available for this project)

We want to maximize her profit, which is given by:
P = 80y + 40x

We can use linear programming to solve this problem. Here's the set up:

Maximize P = 80y + 40x

Subject to:
x + y ≤ 16
y ≤ 12
50x + 20y ≤ 500

We can graph these inequalities and find the feasible region:

<img src="https://i.imgur.com/m7mNvap.png" alt="Feasible region graph">

The maximum profit will occur at one of the corner points of the feasible region. We can evaluate P at each corner point:

- (0, 12): P = 80(12) + 40(0) = 960
- (8, 8): P = 80(8) + 40(8) = 960
- (10, 6): P = 80(6) + 40(10) = 920
- (16, 0): P = 80(0) + 40(16) = 640

So the maximum profit the farmer can make is GH₵960.00 if she has 8 pigs and 8 goats.