1. **Problem Statement:** We have two cylinders: - Cylinder A: height = $x$, diameter = $y$ - Cylinder B: height = $y$, diameter = $x$ Lucy claims that if $y > x > 0$, then the volume of Cylinder A (diameter $y$) is greater than Cylinder B (height $y$). 2. **Volume Formula for a Cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Calculate Volume of Cylinder A:** Radius $r = \frac{y}{2}$, height $h = x$ $$V_A = \pi \left(\frac{y}{2}\right)^2 x = \pi \frac{y^2}{4} x = \frac{\pi}{4} y^2 x$$ 4. **Calculate Volume of Cylinder B:** Radius $r = \frac{x}{2}$, height $h = y$ $$V_B = \pi \left(\frac{x}{2}\right)^2 y = \pi \frac{x^2}{4} y = \frac{\pi}{4} x^2 y$$ 5. **Compare Volumes:** Since $\frac{\pi}{4}$ and $y$ are positive, compare $y^2 x$ and $x^2 y$: $$y^2 x \quad \text{vs} \quad x^2 y$$ Divide both sides by $xy$ (positive, so inequality direction preserved): $$\frac{y^2 x}{xy} = y \quad \text{and} \quad \frac{x^2 y}{xy} = x$$ Since $y > x$, we have: $$y > x$$ Therefore, $$V_A > V_B$$ 6. **Conclusion on Lucy's Proof:** Lucy's algebraic steps are correct and her conclusion that the cylinder with diameter $y$ has greater volume when $y > x > 0$ is valid. **Final answer:** Lucy's claim is correct, and her proof is valid.