The demand function for a commodity is \(p = 35 - 2x - x^2\), then the consumer's surplus at equilibrium price \(p_0 = 20\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 27
At the equilibrium price \(p_0 = 20\), the quantity demanded satisfies
\(35 - 2x - x^2 = 20\)
\(x^2 + 2x - 15 = 0\)
\((x + 5)(x - 3) = 0\)
Quantity cannot be negative, so \(x_0 = 3\).
The consumer’s surplus is the area under the demand curve up to \(x_0\), minus the amount actually spent, \(p_0x_0\):
\( \displaystyle CS = \int_{0}^{3} (35 - 2x - x^2)\,dx - 20 \times 3 \)
\( \displaystyle = \left[35x - x^2 - \frac{x^3}{3}\right]_0^3 - 60 \)
\(= (105 - 9 - 9) - 60\)
\(= 87 - 60 = 27\)
CBSE Applied Mathematics (Class XII):
Consumers Surplus and Producers Surplus
