Question 2 OR (a)

Profit-maximising output from a demand curve and an average cost curve

9 marks

Time to write: About 18 minutes
Topic: Maxima and minima in one variable

Question

Question 2 OR (a) · 9 marks

A firm faces the demand curve x = 240 - 3 p and the average cost curve AC = 10 + x/60, where p denotes the price per kg and x the output (in kg). Determine the output that gives the maximum profit. Calculate the corresponding maximum profit.

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

18 minfor 9 marks

Read from the question

Determine…Calculate…Two asks in one question: answer both.
DetermineFormula first, then the working, then the answer with its unit (₹, %, units).

Write it in this order

Read the data; note exactly what is asked 1 min
1 Revenue, cost and profit functions 4 marks 7 min
2 Profit-maximising output with second-order condition 3 marks 5 min
3 Maximum profit 2 marks 4 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

123

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Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Demand x = 240 − 3p gives p = 80 − x/3. Average cost AC = 10 + x/60, so total cost TC = x·AC = 10x + x²/60.

Profit function. TR = px = 80x − x²/3. Profit π = TR − TC:

π = 80x − x²/3 − 10x − x²/60 = 70x − (21/60)x² = 70x − 0.35x².

Maximisation. dπ/dx = 70 − 0.7x = 0 ⟹ x = 100 kg; d²π/dx² = −0.7 < 0, so profit is a maximum. (At x = 100: MR = 80 − 2x/3 = 13.33 and MC = 10 + x/30 = 13.33.)

Maximum profit

At x = 100 kg	Working	₹
Price	80 − 100/3	46.67 per kg
Total revenue	80(100) − 100²/3	4,666.67
Total cost	10(100) + 100²/60	1,166.67
Maximum profit	70(100) − 0.35(100)²	3,500.00

Output 100 kg gives the maximum profit of ₹3,500 (price ₹46.67 a kg, average cost ₹11.67 a kg).

Where the 9 marks sit

#	Component	Marks
1	Revenue, cost and profit functions	4
2	Profit-maximising output with second-order condition	3
3	Maximum profit	2
	Total	9

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