The problem asks for the perimeter of the loop of the curve defined by the equation:

$3a y^2 = x(x - 2)^2$

The solution provided is:

$\text{Perimeter} = \frac{4a}{\sqrt{3}} \, \text{units}$

How to Approach:

This type of problem involves parametric equations, calculus, and arc length to calculate the perimeter of a loop of a given curve. Let me outline the general steps:

1. Equation Analysis: The given equation is $3a y^2 = x(x-2)^2$. This is a quartic curve, meaning it has a degree of 4 in $x$. To understand the loop, we need to analyze this curve's behavior, specifically finding where it forms a closed loop.

2. Identifying the Loop: We solve for points where the loop occurs by looking for values of $x$ and $y$ that define a finite region. Typically, we check where $y = 0$ and find the relevant $x$-values to mark the boundary of the loop.

3. Arc Length Formula: The perimeter (arc length) of a curve is given by the integral: $P = \int_{x_1}^{x_2} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$ where $x_1$ and $x_2$ are the bounds of the loop, and $y$ is a function of $x$.

4. Differentiating the Equation: To apply the arc length formula, we would implicitly differentiate the given equation to find $\frac{dy}{dx}$ and substitute it into the arc length formula.

5. Solving the Integral: Finally, we compute the integral using the bounds of the loop. In this case, the answer simplifies to $\frac{4a}{\sqrt{3}}$, meaning the integral likely involves simplification techniques or known results for similar curves.

Would you like me to dive deeper into any of these steps or provide additional details on the derivation?

Follow-up Questions:

1. How can we differentiate implicit equations like $3a y^2 = x(x-2)^2$?
2. What technique can simplify the arc length integral for this curve?
3. How can we determine the exact bounds of the loop on the $x$-axis?
4. Why does the perimeter simplify to a formula like $\frac{4a}{\sqrt{3}}$?
5. Can we find the area enclosed by the loop as well?

Tip:

When dealing with arc length, always ensure you have correctly differentiated the curve's equation and carefully set the integral bounds based on the region of interest.