Key Concepts and Formulas

• Area of a Triangle: Given two sides $a$ and $b$ and the included angle $C$, the area is given by $Area = \frac{1}{2}ab\sin C$.
• Maximum Value of Sine: The maximum value of $\sin \theta$ is 1, which occurs when $\theta = 90^\circ$ or $\frac{\pi}{2}$ radians.

Step-by-Step Solution

Step 1: Define Variables and Express the Area

We are given a triangle with two sides of length $x$ and the angle between them, $\theta$. The third side is along a riverbank. We want to maximize the area. Let the two sides with length $x$ be $AB$ and $AC$, such that $AB = AC = x$, and $\angle BAC = \theta$. The area of the triangle $ABC$ can be expressed using the formula:

$A = \frac{1}{2} \cdot AB \cdot AC \cdot \sin{\theta} = \frac{1}{2} \cdot x \cdot x \cdot \sin{\theta} = \frac{1}{2}x^2\sin{\theta}$

Explanation: We use the formula for the area of a triangle given two sides and the included angle. This allows us to express the area as a function of $\theta$, which is the variable we will optimize.

Step 2: Maximize the Area with Respect to $\theta$

The area $A$ is given by $A = \frac{1}{2}x^2\sin{\theta}$. Since $x$ is a constant, we need to maximize $\sin{\theta}$ to maximize $A$. The maximum value of $\sin{\theta}$ is 1, which occurs when $\theta = 90^\circ$ or $\frac{\pi}{2}$ radians. Since $\theta$ is an angle in a triangle, it must be between $0$ and $\pi$. Therefore, $\theta = \frac{\pi}{2}$ is a valid angle.

Explanation: We identify that the area is directly proportional to $\sin{\theta}$. Therefore, maximizing the area is equivalent to maximizing $\sin{\theta}$. We know the maximum value of sine is 1, so we find the angle at which this occurs.

Step 3: Calculate the Maximum Area

Substitute the maximum value of $\sin{\theta}$ (which is 1) into the area formula:

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

Explanation: We substitute the value of $\sin{\theta}$ that maximizes the area into the area formula to find the maximum possible area.

Common Mistakes & Tips

• Using the wrong formula: Choosing the wrong area formula can make the problem more difficult. Using the formula involving two sides and the included angle is the most straightforward approach here.
• Forgetting the range of $\theta$: The angle $\theta$ must be between 0 and $\pi$ for a valid triangle.

Summary

The area of the triangular park is given by $A = \frac{1}{2}x^2\sin{\theta}$. To maximize the area, we need to maximize $\sin{\theta}$, which has a maximum value of 1 when $\theta = 90^{\circ}$. Substituting this into the area formula, we find that the maximum area is $\frac{1}{2}x^2$.

The final answer is $\boxed{{1 \over 2}{x^2}}$, which corresponds to option (C).