1. **Stating the problem:** We have a cone with height 120 cm and base diameter 80 cm (thus radius 40 cm). Inside it, a cylinder is inscribed with height $h$ and radius $r$. We want to find the relationship between $h$ and $r$. 2. **Understanding the geometry:** The cone's height is 120 cm and radius is 40 cm. The cylinder fits inside the cone, so its radius $r$ and height $h$ must satisfy the cone's linear profile. 3. **Using similar triangles:** The radius of the cone decreases linearly from 40 cm at the base ($h=0$) to 0 at the apex ($h=120$). The radius at height $h$ is proportional to the remaining height of the cone above the cylinder. 4. **Setting up the relation:** The radius at height $h$ is $r$, and the height from the apex to the top of the cylinder is $120 - h$. By similar triangles: $$\frac{r}{40} = \frac{120 - h}{120}$$ 5. **Solving for $h$ in terms of $r$:** Multiply both sides by 120: $$120 \cdot \frac{r}{40} = 120 - h$$ Simplify: $$3r = 120 - h$$ Rearranged: $$h = 120 - 3r$$ 6. **Interpretation:** This formula shows that as the radius $r$ of the inscribed cylinder increases, its height $h$ decreases linearly. **Final answer:** $$h = 120 - 3r$$