problem: One day, I decide to run to the park. On the way there, I run at a rate of $x^2$ miles per hour for $3$ hours. On the way back, I take the same path and jog at a slower rate of $16 - 4x$ miles per hour so that it takes me $4$ hours to get home. Given that $x > 0$, what is $x$? Express your answer as a common fraction.
solution: Since we know that the distance that I traveled to the park and the distance that I traveled to get back home are the same, and that distance $=$ rate $\times$ time, we have that  \begin{align*}
  & (x^2)(3) = (16 - 4x)(4) \\
  \Rightarrow\qquad & 3x^2 = 64 - 16x \\
  \Rightarrow\qquad & 3x^2 + 16x - 64 = 0 \\
  \Rightarrow\qquad & (3x - 8)(x + 8) = 0.
  \end{align*}Solving this equation, we get the solutions $x = \frac{8}{3}$ and $x = -8$. Since $x$ must be positive, we have $x = \boxed{\frac{8}{3}}$.