The total revenue and the total cost (in Rs.) of producing and selling x units of a product are \(R(x) = 90x - x^2\) and \(C(x) = x^2 + 10x + 200\). Consider the following statements:
(A) The profit is maximum when 20 units are produced.
(B) The maximum profit is Rs. 600.
(C) At the output of maximum profit, the marginal revenue equals the marginal cost.
(D) The marginal cost at x = 20 is Rs. 40.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
The profit is revenue minus cost:
\(P(x) = 90x - x^2 - x^2 - 10x - 200\)
\(P(x) = -2x^2 + 80x - 200\)
\(P'(x) = -4x + 80 = 0 \ \Rightarrow \ x = 20\)
\(P''(x) = -4 \lt 0\)
\(P(20) = -800 + 1600 - 200 = 600\)
The marginal revenue and marginal cost at x = 20:
\(MR = 90 - 2x = 50\)
\(MC = 2x + 10 = 50\)
So the profit is maximum at x = 20, the maximum profit is Rs. 600, and MR = MC = Rs. 50 at x = 20.
Checking the statements
(A) The profit is maximum when 20 units are produced. — Correct. P′(20) = 0 and P″(x) = −4 < 0.
(B) The maximum profit is Rs. 600. — Correct. P(20) = −800 + 1600 − 200 = 600.
(C) At the output of maximum profit, the marginal revenue equals the marginal cost. — Correct. At x = 20, MR = 90 − 40 = 50 and MC = 40 + 10 = 50.
(D) The marginal cost at x = 20 is Rs. 40. — Incorrect. MC = 2(20) + 10 = 50.