1. **State the problem:** We are told that $y$ is directly proportional to $x^3$, and that $y=20h$ when $h=x$. We need to find $y$ in terms of $h$ and $x$. 2. **Write the proportionality formula:** Since $y$ is directly proportional to $x^3$, we write: $$y = kx^3$$ where $k$ is the constant of proportionality. 3. **Use the given condition to find $k$:** When $h = x$, we have $y = 20h$. Substitute $h = x$ into this: $$y = 20x$$ But from the proportionality, $y = kx^3$. So: $$kx^3 = 20x$$ 4. **Solve for $k$:** Divide both sides by $x$ (assuming $x \neq 0$): $$kx^2 = 20$$ Therefore, $$k = \frac{20}{x^2}$$ 5. **Write $y$ in terms of $h$ and $x$:** Since $y = kx^3$, substitute $k$: $$y = \frac{20}{x^2} x^3 = 20x$$ But we want $y$ in terms of $h$ and $x$. Recall from the problem that $y=20h$ when $h=x$, so the general form relating $y$ and $h$ is: Since $y$ is proportional to $x^3$, and $y=20h$ when $h=x$, we can express $y$ as: $$y = 20 \frac{h}{x} x^3 = 20 h x^2$$ **Final answer:** $$y = 20 h x^2$$