The demand function of a product is \(p = 60 - 3x\), where p is the price (in Rs.) per unit and x is the number of units demanded. Consider the following statements:
(A) The average revenue when 5 units are sold is Rs. 45.
(B) The marginal revenue when 5 units are sold is Rs. 15.
(C) The total revenue is maximum when 10 units are sold.
(D) The maximum total revenue is Rs. 300.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
The total revenue is price × quantity:
\(R(x) = px = 60x - 3x^2\)
\( \displaystyle AR = \frac{R(x)}{x} = 60 - 3x \)
\( \displaystyle MR = \frac{dR}{dx} = 60 - 6x \)
R is maximum where MR = 0, that is x = 10 (and \(R''(x) = -6 \lt 0\)):
\(R(10) = 600 - 300 = 300\)
So AR(5) = 45, MR(5) = 30, and the maximum revenue is Rs. 300 at x = 10.
Checking the statements
(A) The average revenue when 5 units are sold is Rs. 45. — Correct. AR(5) = 60 − 15 = 45.
(B) The marginal revenue when 5 units are sold is Rs. 15. — Incorrect. MR(5) = 60 − 30 = 30.
(C) The total revenue is maximum when 10 units are sold. — Correct. MR = 60 − 6x = 0 at x = 10, and R′′ = −6 < 0.
(D) The maximum total revenue is Rs. 300. — Correct. R(10) = 600 − 300 = 300.