The demand function for a commodity is $p = 35 - 2x - x^2$, then the consumer's surplus at equilibrium price $p_0 = 20$ is

The equilibrium price is $p_0 = 20$.

The demand function is $p = 35 - 2x - x^2$.

Setting $p = 20$:

$20 = 35 - 2x - x^2$

$x^2 + 2x - 15 = 0$

$(x + 5)(x - 3) = 0$

This gives $x = -5$ or $x = 3$.

Since quantity cannot be negative, $x_0 = 3$.

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Consumer's surplus is given by:

$CS = \int_0^{x_0} (35 - 2x - x^2) \, dx - p_0 \cdot x_0$

$CS = \int_0^3 (35 - 2x - x^2) \, dx - 20 \times 3$

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Evaluating the integral:

$\int_0^3 (35 - 2x - x^2) \, dx = \left[35x - x^2 - \frac{x^3}{3}\right]_0^3$

At $x = 3$:

$= 35(3) - (3)^2 - \frac{(3)^3}{3}$

$= 105 - 9 - 9$

$= 87$

At $x = 0$: the value is $0$.

Therefore, $\int_0^3 (35 - 2x - x^2) \, dx = 87$

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Consumer's surplus:

$CS = 87 - 60$

$CS = 27$

Therefore, the consumer's surplus at equilibrium price $p_0 = 20$ is $27$.