1. **Problem statement:** A manufacturer wants to minimize the cost of producing a cylindrical can with volume 340 cm³. The cost per cm² for the top and bottom discs is 0.1 cents. The cost per cm² for the curved surface is 0.05 cents. We need to express the total cost $C$ in terms of the radius $r$. 2. **Known formulas:** - Volume of cylinder: $$V = \pi r^2 h$$ - Surface areas: - Top and bottom discs area: $$2 \pi r^2$$ - Curved surface area: $$2 \pi r h$$ 3. **Given:** $$V = 340 = \pi r^2 h \implies h = \frac{340}{\pi r^2}$$ 4. **Cost calculation:** - Cost of top and bottom: $$0.1 \times 2 \pi r^2 = 0.2 \pi r^2$$ - Cost of curved surface: $$0.05 \times 2 \pi r h = 0.1 \pi r h$$ 5. **Substitute $h$ into curved surface cost:** $$0.1 \pi r \times \frac{340}{\pi r^2} = 0.1 \pi r \times \frac{340}{\pi r^2} = 0.1 \times 340 \times \frac{1}{r} = \frac{34}{r}$$ 6. **Total cost $C$:** $$C = 0.2 \pi r^2 + \frac{34}{r}$$ 7. **Interpretation:** The cost function in terms of $r$ is $$C = \frac{34}{r} + 0.2 \pi r^2$$ This matches the first option given. **Final answer:** $$C = \frac{34}{r} + 0.2 \pi r^2$$