Question

A farmer wants to sell his produce in two neighboring towns, A and B. He can grow up to 700 pounds of vegetables, but the demand for vegetables is at most 400 pounds in Town A, where he sells his produce at $3.25 per pound, and 600 pounds in Town B, where the price is $2.99 per pound. How much should the farmer sell in each town to maximize his profits?

a) Sell 300 pounds in Town A and 400 pounds in Town B
b) Sell 400 pounds in Town A and 300 pounds in Town B
c) Sell 200 pounds in Town A and 500 pounds in Town B
d) Sell 500 pounds in Town A and 200 pounds in Town B

Answer 1

To maximize profits, the farmer should sell 200 pounds in Town A and 500 pounds in Town B.

Step-by-step explanation:

To maximize profits, the farmer should find the optimal quantity to sell in each town. Let's assume the farmer sells x pounds in Town A and y pounds in Town B.

The total amount of vegetables the farmer can grow is given by the equation x + y = 700 (since he can grow up to 700 pounds).

To determine the optimal quantity, we need to consider the demand and price in each town. In Town A, the demand is at most 400 pounds, so the farmer can sell x = 400 pounds at a price of $3.25 per pound. In Town B, the demand is at most 600 pounds, so the farmer can sell y = 600 pounds at a price of $2.99 per pound.

Using these conditions, we can solve the system of equations:

x + y = 700 (equation 1)

x ≤ 400 (equation 2)

y ≤ 600 (equation 3)

Solving the system of equations, we find that the solution is option c) Sell 200 pounds in Town A and 500 pounds in Town B.