The demand function $P$ for maximising a profit monopolist is given by $P=274-x^2$, while the marginal cost is $4+3x$ for $x$ units of commodity. The consumer surplus is

486

The correct answer is Option (4) → 486

Given demand function: $P = 274 - x^2$

Marginal cost: $MC = 4 + 3x$

For monopoly, profit is maximised where $MR = MC$.

Total Revenue: $TR = P \cdot x = (274 - x^2)x = 274x - x^3$

Marginal Revenue: $MR = \frac{d(TR)}{dx} = 274 - 3x^2$

Equilibrium condition: $MR = MC$

$274 - 3x^2 = 4 + 3x$

$274 - 4 = 3x^2 + 3x$

$270 = 3x^2 + 3x$

$x^2 + x - 90 = 0$

$(x+10)(x-9) = 0$

$x = 9$ (positive root)

Corresponding price: $P = 274 - (9)^2 = 274 - 81 = 193$

Consumer surplus formula:

$CS = \int_0^{x_0} P(x)\,dx - P(x_0)\cdot x_0$

$CS = \int_0^9 (274 - x^2)\,dx - (193)(9)$

$\int_0^9 (274 - x^2)\,dx = [274x - \frac{x^3}{3}]_0^9$

$= (274\cdot 9 - \frac{729}{3}) - 0$

$= 2466 - 243 = 2223$

Now subtract: $P(9)\cdot 9 = 193 \cdot 9 = 1737$

$CS = 2223 - 1737 = 486$

Final Answer: Consumer surplus = $486$