Question

Profit from Farm Animals Jan raises only pigs and geese. She wants to raise no more than 16 animals, with no more than 12 geese. She spends $50 to raise a pig and $20 to raise a goose. She has $500 available for this purpose. Find the maximum profit she can make if she makes a profit of $80 per goose and $40 per pig, and determine how many pigs and geese she should raise to achieve this maximum.

Answer 1

This is a linear programming program in which you want to create a system of inequalities and optimize the solution.

We want to maximize the profit, so we want to create one equation that represents the profit, and optimize that function. let's let g = # of geese and p = # of pigs. since she makes $80 profit per goose, and $40 per pig, the optimization function is

profit = 80g + 40p

now, let's create inequalities based on the constraints

She wants to raise no more than 16 animals ---> p + g ≤ 16

with no more than 12 geese ---> g ≤ 12

She spends $50 to raise a pig and $20 to raise a goose. She has $500 available for this purpose. ---> she spends 50 per pig, so 50p, and similarly, 20 per goose is 20g. she only has 500 to spend, so 50p + 20g ≤ 500

now we have our system of inequalities. graph the system, shade the area where all the inequalies overlap. at each vertex of the shaded area is a potential point at which the optimization function is optimized. check each vertex by plugging it into the optimization function, then see which p and g value gives you the largest profit value.

Answer 2

@tranquility wondering if you could take a look and make sure I set this up correctly? when you get a chance.

Answer 3

It looks correct to me! The only other constraint missing would be
\( p \ge 0\)
\( g \ge 0\)