Given that $y = \frac{4}{5}\left(\frac{x}{12} - 1\right)^6$ and that both $x$ and $y$ vary with time, find the value of $y$ when the rate of change of $y$ is $12\frac{4}{5}$ times the rate of change of $x$.

1. Find the derivative of $y$ with respect to $x$.

$y = \frac{4}{5}\left(\frac{x}{12} - 1\right)^6$

$\frac{dy}{dx} = \frac{4}{5} \times 6\left(\frac{x}{12} - 1\right)^5 \times \frac{1}{12}$

$\frac{dy}{dx} = \frac{2}{5}\left(\frac{x}{12} - 1\right)^5$

2. Use the given rate of change relationship to find the value of $\frac{dy}{dx}$.

Given $\frac{dy}{dt} = 12\frac{4}{5} \frac{dx}{dt} = \frac{64}{5} \frac{dx}{dt}$.

By the chain rule, $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$.

Therefore, $\frac{dy}{dx} = \frac{64}{5}$.

3. Equate the derivative to $\frac{64}{5}$ and solve for the inner expression.

$\frac{2}{5}\left(\frac{x}{12} - 1\right)^5 = \frac{64}{5}$

$\left(\frac{x}{12} - 1\right)^5 = 32$

$\frac{x}{12} - 1 = 2$

4. Substitute the value of the inner expression back into the original equation to find $y$.

Substitute $\left(\frac{x}{12} - 1\right) = 2$ into the equation for $y$:

$y = \frac{4}{5}(2)^6$

$y = \frac{4}{5}(64) = \frac{256}{5} = 51\frac{1}{5}$