Key Concepts and Formulas

• Parametric Differentiation: If $x = f(t)$ and $y = g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.
• Slope of Tangent: The slope of the tangent to a curve at a point is given by $\frac{dy}{dx}$, which is also equal to $\tan \theta$, where $\theta$ is the angle the tangent makes with the positive x-axis.
• Trigonometric Identities: $1 + \cos 2t = 2\cos^2 t$, $\sin(2t) = 2\sin t \cos t$.

Step-by-Step Solution

Step 1: Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$

We are given $x = 12(t + \sin t \cos t)$ and $y = 12(1 + \sin t)^2$. Our goal is to find the derivatives of $x$ and $y$ with respect to $t$.

$x = 12(t + \sin t \cos t)$ $\frac{dx}{dt} = 12\left(1 + \cos^2 t - \sin^2 t\right) = 12(1 + \cos 2t) = 12(2\cos^2 t) = 24\cos^2 t$

$y = 12(1 + \sin t)^2$ $\frac{dy}{dt} = 12 \cdot 2(1 + \sin t) \cdot \cos t = 24(1 + \sin t)\cos t$

Step 2: Calculate $\frac{dy}{dx}$

Now, we compute the derivative $\frac{dy}{dx}$ using the parametric differentiation formula: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{24(1 + \sin t)\cos t}{24\cos^2 t} = \frac{1 + \sin t}{\cos t}$

Step 3: Simplify $\frac{dy}{dx}$ and relate to the given angle

We are given that the tangent makes an angle of $\frac{\pi}{3}$ with the positive x-axis. Therefore, $\frac{dy}{dx} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}$. Hence, $\frac{1 + \sin t}{\cos t} = \sqrt{3}$ $1 + \sin t = \sqrt{3}\cos t$ $\sin t - \sqrt{3}\cos t = -1$ Divide by 2: $\frac{1}{2}\sin t - \frac{\sqrt{3}}{2}\cos t = -\frac{1}{2}$ $\sin t \cos\left(\frac{\pi}{3}\right) - \cos t \sin\left(\frac{\pi}{3}\right) = -\frac{1}{2}$ $\sin\left(t - \frac{\pi}{3}\right) = -\frac{1}{2}$ Since $0 < t < \frac{\pi}{2}$, we have $-\frac{\pi}{3} < t - \frac{\pi}{3} < \frac{\pi}{6}$. Thus, $t - \frac{\pi}{3} = -\frac{\pi}{6}$ $t = \frac{\pi}{3} - \frac{\pi}{6} = \frac{2\pi - \pi}{6} = \frac{\pi}{6}$

Step 4: Find $y_0$

We need to find $y_0$, which is the value of $y$ when $t = \frac{\pi}{6}$. $y_0 = 12\left(1 + \sin\left(\frac{\pi}{6}\right)\right)^2 = 12\left(1 + \frac{1}{2}\right)^2 = 12\left(\frac{3}{2}\right)^2 = 12\left(\frac{9}{4}\right) = 3 \cdot 9 = 27$

Step 5: Find the correct option

The problem statement in the prompt is incorrect. After careful calculations, the value of $y_0$ is found to be 27.

Common Mistakes & Tips

• Trigonometric Errors: Be extremely careful when applying trigonometric identities. Double-check each substitution and simplification.
• Domain Check: Always verify that the value of $t$ you find lies within the given domain.
• Simplification: Simplifying the expression for $\frac{dy}{dx}$ before substituting the angle value can save time and reduce errors.

Summary

We found $\frac{dy}{dx}$ using parametric differentiation. By setting this equal to $\tan(\frac{\pi}{3})$, we found that $t = \frac{\pi}{6}$. Substituting this value of $t$ into the equation for $y$ gives $y_0 = 27$.

The final answer is $\boxed{27}$, which corresponds to option (C).