Problem Explanation:

We need to find the ratio of the height ($h$) to the radius ($r$) of a cone such that the surface area of the cone is minimized for a fixed volume.

Solution:

1. Volume of the Cone: The volume $V$ of a cone is given by: $V = \frac{1}{3} \pi r^2 h$ Since $V$ is fixed, this equation relates $h$ and $r$.

2. Surface Area of the Cone: The surface area $S$ of the cone (excluding the base) is: $S = \pi r l$ where $l$ is the slant height of the cone. From the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$ Substituting for $l$, the surface area becomes: $S = \pi r \sqrt{r^2 + h^2}$

3. Express $h$ in Terms of $r$: From the volume equation: $h = \frac{3V}{\pi r^2}$

4. Substitute $h$ into $S$: Substituting $h = \frac{3V}{\pi r^2}$ into $S = \pi r \sqrt{r^2 + h^2}$: $S = \pi r \sqrt{r^2 + \left(\frac{3V}{\pi r^2}\right)^2}$

5. Simplify the Surface Area Expression: Let $k = \frac{3V}{\pi}$. Then: $h = \frac{k}{r^2}, \quad h^2 = \frac{k^2}{r^4}$ Substituting into $S$: $S = \pi r \sqrt{r^2 + \frac{k^2}{r^4}}$ $S = \pi r \sqrt{\frac{r^6 + k^2}{r^4}}$ $S = \pi \sqrt{r^6 + k^2} / r$

6. Minimize $S$: To minimize $S$, differentiate $S$ with respect to $r$ and set the derivative to zero. Using $S = \pi r \sqrt{r^2 + \frac{k^2}{r^4}}$: $\frac{dS}{dr} = \pi \left( \sqrt{r^2 + \frac{k^2}{r^4}} + r \cdot \frac{1}{2\sqrt{r^2 + \frac{k^2}{r^4}}} \cdot \left( 2r - \frac{4k^2}{r^5} \right) \right)$ Setting $\frac{dS}{dr} = 0$, solve for $r$ and $h$. This results in the ratio: $\frac{h}{r} = \sqrt{2}$

Final Answer:

The ratio of the height to the radius for minimizing the material used is: $\boxed{\sqrt{2}}$

Would you like further clarification or step-by-step details on any part?

Related Questions:

1. How does the volume equation relate to the optimization problem?
2. Can this optimization method be extended to other geometric shapes?
3. How does the ratio $\sqrt{2}$ affect the physical design of the cone?
4. What role does the slant height $l$ play in the surface area calculation?
5. How would the solution differ if the cone included the base in its surface area?

Tip:

When optimizing shapes for fixed constraints (like volume), always express one variable in terms of the other using the constraint equation before differentiating.