To minimize the surface area of an open-top cylindrical container with a fixed volume, we can follow these steps:

1. Define the Variables and Equations:

   • Let $r$ be the radius of the cylinder's base, and $h$ be its height.
   • The volume $V$ is given by: $V = \pi r^2 h$
   • Since the volume is given as $V = 2197 \, \text{cm}^3$, we have: $2197 = \pi r^2 h$
   • The surface area $S$ for an open-top cylinder is: $S = \pi r^2 + 2 \pi r h$ where $\pi r^2$ is the area of the base and $2 \pi r h$ is the area of the side.

2. Express $h$ in Terms of $r$ Using the Volume: $h = \frac{2197}{\pi r^2}$

3. Substitute $h$ in the Surface Area Formula: Substitute $h$ from the volume equation into the surface area formula: $S = \pi r^2 + 2 \pi r \left( \frac{2197}{\pi r^2} \right)$ Simplifying, we get: $S = \pi r^2 + \frac{4394}{r}$

4. Minimize $S$ by Taking the Derivative with Respect to $r$: Differentiate $S$ with respect to $r$ and set it to zero to find the critical points: $\frac{dS}{dr} = 2 \pi r - \frac{4394}{r^2} = 0$ Rearranging, we get: $2 \pi r^3 = 4394$ $r^3 = \frac{4394}{2 \pi}$ Solving for $r$: $r = \sqrt[3]{\frac{4394}{2 \pi}}$

5. Calculate $r$ and $h$: Using this value of $r$, calculate $h$ using $h = \frac{2197}{\pi r^2}$.

Let's compute these values:The dimensions that will minimize the surface area are approximately:

• Radius (r): 8.88 cm
• Height (h): 8.88 cm

This symmetry between the radius and height results from minimizing the surface area for a given volume in an open-top cylindrical container.

Would you like more details on this solution, or have any questions?

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Here are some additional questions you might explore:

1. How would the dimensions change if the container had a closed top?
2. How does the surface area change with small increases or decreases in volume?
3. Could you use a different shape to hold the same volume with less surface area?
4. What would be the surface area of this cylinder with the optimal dimensions?
5. How does the optimization differ if the radius is fixed and height is variable?

Tip: In calculus, minimizing surface area with volume constraints often results in symmetrical dimensions, especially in shapes like cylinders.