The demand function P for maximising a profit monopolist is given by P=274-x², while the marginal cost is 4+3x for x units of commodity. The consumer surplus is

The demand function is $P = 274 - x^2$ and the marginal cost is $MC = 4 + 3x$.

A monopolist maximizes profit where Marginal Revenue equals Marginal Cost.

Total Revenue is:

$TR = P \times x$

$TR = (274 - x^2) \times x$

$TR = 274x - x^3$

Marginal Revenue is the derivative of Total Revenue:

$MR = \frac{d(TR)}{dx}$

$MR = 274 - 3x^2$

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Setting $MR = MC$:

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

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

$270 = 3x^2 + 3x$

$90 = x^2 + x$

$x^2 + x - 90 = 0$

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

Taking the positive value: $x = 9$

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The price at $x = 9$:

$P = 274 - x^2$

$P = 274 - (9)^2$

$P = 274 - 81$

$P = 193$

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Consumer Surplus is the area between the demand curve and the price level:

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

Evaluating the integral:

$\int (274 - x^2) dx = 274x - \frac{x^3}{3}$

Applying limits from 0 to 9:

$= \left[274(9) - \frac{(9)^3}{3}\right] - \left[274(0) - \frac{(0)^3}{3}\right]$

$= 2466 - \frac{729}{3}$

$= 2466 - 243$

$= 2223$

Therefore:

$CS = 2223 - (193 \times 9)$

$CS = 2223 - 1737$

$CS = 486$

The consumer surplus is $486$.