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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.
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