1. Interpret the geometry

A triangular park is formed with:

• two equal fenced sides, each of length $x$,
• the third side along the river bank.

So the triangle is an isosceles triangle with two equal sides $x$ and included angle $\theta$ between them.

2. Write the area in terms of $\theta$

For a triangle with two sides $a,b$ and included angle $\theta$,

$A=\frac12 ab\sin\theta$

Here $a=b=x$, so

$A=\frac12 x\cdot x\sin\theta=\frac12 x^2\sin\theta$

3. Maximize the area

Since $x$ is fixed, maximizing $A$ means maximizing $\sin\theta$.

We know

$\sin\theta \le 1$

with equality when

$\theta=90^\circ$

Hence the maximum area is

$A_{\max}=\frac12 x^2\cdot 1=\frac12 x^2$

4. Check the options

• A: $\dfrac32 x^2$ — too large, not possible.
• B: $\sqrt{\dfrac{x^3}{8}}$ — dimensionally incorrect for area.
• C: $\dfrac12 x^2$ — correct.
• D: $\pi x^2$ — not related to triangle area here.

Therefore, the correct option is C.