1. **Stating the problem:** We are given that $y$ varies directly as the cube root of $x$, i.e., $y \propto \sqrt[3]{x}$. We need to find the constant of proportionality $k$ and then use it to find $x$ when $y=2.4$. 2. **Formula and rules:** Since $y$ varies directly as $\sqrt[3]{x}$, the formula is: $$y = k \sqrt[3]{x}$$ where $k$ is a constant. 3. **Find $k$ using given values:** Given $y=3$ when $x=125$: $$3 = k \sqrt[3]{125}$$ Since $\sqrt[3]{125} = 5$, we have: $$3 = 5k$$ Dividing both sides by 5: $$3 = \cancel{5}k / \cancel{5}$$ $$k = \frac{3}{5}$$ 4. **Find $x$ when $y=2.4$:** Using $y = k \sqrt[3]{x}$: $$2.4 = \frac{3}{5} \sqrt[3]{x}$$ Multiply both sides by $\frac{5}{3}$: $$2.4 \times \frac{5}{3} = \cancel{\frac{3}{5}} \sqrt[3]{x} \times \cancel{\frac{5}{3}}$$ $$4 = \sqrt[3]{x}$$ 5. **Solve for $x$ by cubing both sides:** $$x = 4^3 = 64$$ **Final answer:** $$x = 64$$