If under pure competition demand and supply functions are given by $p = \sqrt{10-x}$ and $p =\frac{1}{2} (x-2)$ respectively, where p is price per unit and x is quantity, then the consumer surplus is:

$\frac{2}{3} [10\sqrt{10}-26]$

The correct answer is Option (3) → $\frac{2}{3} [10\sqrt{10}-26]$ **

Demand function: $p=\sqrt{10-x}$

Supply function: $p=\frac{1}{2}(x-2)$

Step 1: Find equilibrium quantity $x_e$

Set demand = supply:

$\sqrt{10-x}=\frac{1}{2}(x-2)$

Square both sides:

$10-x=\frac{1}{4}(x-2)^{2}$

$40-4x=(x-2)^{2}$

$40-4x=x^{2}-4x+4$

$x^{2}-4x+4 = 40-4x$

$x^{2}-36=0$

$x=\sqrt{36}=6$ (positive quantity)

Equilibrium price:

$p_e=\frac{1}{2}(6-2)=2$

Step 2: Consumer Surplus

CS = $\displaystyle \int_{0}^{6} \sqrt{10-x}\,dx \;-\; (p_e \cdot x_e)$

Compute integral:

Let $I=\int_{0}^{6} \sqrt{10-x}\,dx$

Use substitution: $u=10-x,\; du=-dx$

When $x=0,\; u=10$

When $x=6,\; u=4$

$I=\int_{10}^{4} \sqrt{u}\,(-du)=\int_{4}^{10} u^{1/2}\,du$

$I=\left[\frac{2}{3}u^{3/2}\right]_{4}^{10}$

$I=\frac{2}{3}\left(10^{3/2}-4^{3/2}\right)$

$10^{3/2}=10\sqrt{10}$, $4^{3/2}=8$

$I=\frac{2}{3}(10\sqrt{10}-8)$

Subtract equilibrium expenditure:

$p_e x_e = 2 \cdot 6 = 12$

Consumer Surplus:

$\text{CS}=\frac{2}{3}(10\sqrt{10}-8)-12$

Simplify:

$=\frac{20}{3}\sqrt{10}-\frac{16}{3}-12$

$=\frac{20}{3}\sqrt{10}-\frac{16}{3}-\frac{36}{3}$

$=\frac{20}{3}\sqrt{10}-\frac{52}{3}$

Final Answer:

$\displaystyle \text{Consumer Surplus}=\frac{20}{3}\sqrt{10}-\frac{52}{3}$