1. **Problem statement:** We have two similar shapes (cylinders) with given surface areas and volumes. The smaller shape has surface area $110$ mm² and volume $75$ mm³. The larger shape has surface area $350$ mm² and unknown volume. We need to find the volume of the larger shape to 1 decimal place. 2. **Formula and rules:** For similar 3D shapes, the ratio of their surface areas is the square of the scale factor $k$, and the ratio of their volumes is the cube of the scale factor $k$. Let $k$ be the scale factor from the smaller to the larger shape. Surface area ratio: $$\frac{S_2}{S_1} = k^2$$ Volume ratio: $$\frac{V_2}{V_1} = k^3$$ 3. **Calculate the scale factor $k$:** $$k^2 = \frac{350}{110} = \frac{35}{11} \approx 3.1818$$ Taking the square root: $$k = \sqrt{3.1818} \approx 1.783$$ 4. **Calculate the volume of the larger shape:** $$\frac{V_2}{75} = k^3 = (1.783)^3$$ Calculate $k^3$: $$1.783^3 = 1.783 \times 1.783 \times 1.783 \approx 5.67$$ So, $$V_2 = 75 \times 5.67 = 425.25$$ 5. **Final answer rounded to 1 decimal place:** $$\boxed{425.3 \text{ mm}^3}$$