Q1. A company produces two types of widgets, A and B.

The production cost of widget A is $5 per unit and the

selling price is $8 per unit. The production cost of widget B

is $7 per unit and the selling price is $10 per unit. The

company has a fixed cost of $1000 per month and a

production capacity of 500 units per month. How many

units of each type of widget should the company produce to

maximize its profit?

A) 250 units of A and 250 units of B

B) 300 units of A and 200 units of B

C) 200 units of A and 300 units of B

D) 400 units of A and 100 units of B

Answer: C) 200 units of A and 300 units of B

Rationale: The profit function for the company is P = 8A +

10B - 5A - 7B - 1000, where A and B are the number of

units of widget A and B, respectively. To maximize the

profit, we need to find the values of A and B that satisfy the

following conditions:

- The production capacity constraint: A + B <= 500

- The non-negativity constraint: A >= 0 and B >= 0

- The first-order condition: dP/dA = 0 and dP/dB = 0

Solving these equations, we get A = 200 and B = 300.

Therefore, the company should produce 200 units of

widget A and 300 units of widget B to maximize its profit.