1. **Problem statement:** We have two cylinders, a small one with radius $r$ and height $h$, and a large one with radius $2r$ and height $2h$. The total surface area of both cylinders combined is $400\pi$ cm². 2. **Goal:** Express the height $h$ of the small cylinder in terms of the radius $r$. 3. **Surface area formulas:** - Surface area of a cylinder = $2\pi r^2 + 2\pi r h$ (top and bottom circles plus lateral surface). 4. **Calculate surface areas:** - Small cylinder surface area: $A_s = 2\pi r^2 + 2\pi r h$ - Large cylinder surface area: $A_l = 2\pi (2r)^2 + 2\pi (2r)(2h) = 2\pi (4r^2) + 8\pi r h = 8\pi r^2 + 8\pi r h$ 5. **Total surface area:** $$A_s + A_l = (2\pi r^2 + 2\pi r h) + (8\pi r^2 + 8\pi r h) = 10\pi r^2 + 10\pi r h$$ 6. **Given total surface area:** $$10\pi r^2 + 10\pi r h = 400\pi$$ 7. **Divide both sides by $10\pi$:** $$r^2 + r h = 40$$ 8. **Solve for $h$:** $$r h = 40 - r^2$$ $$h = \frac{40 - r^2}{r}$$ **Final answer:** $$h = \frac{40 - r^2}{r}$$