We are given an open-top hollow cylinder, with a volume of $162 \, \text{cm}^3$, and we need to find the minimum surface area.

Formulae involved:

1. Volume of a cylinder: $V = \pi r^2 h$ where:

   • $V$ is the volume,
   • $r$ is the radius,
   • $h$ is the height.

2. Surface area of an open-top cylinder: $A = \pi r^2 + 2\pi r h$ where:

   • The first term $\pi r^2$ is the area of the base,
   • The second term $2\pi r h$ is the lateral surface area.

Step 1: Express height $h$ in terms of $r$ and $V$

From the volume formula: $V = \pi r^2 h \implies h = \frac{V}{\pi r^2}$ Substitute $V = 162 \, \text{cm}^3$: $h = \frac{162}{\pi r^2}$

Step 2: Substitute $h$ into the surface area formula

$A = \pi r^2 + 2\pi r \left( \frac{162}{\pi r^2} \right)$ Simplify: $A = \pi r^2 + \frac{324}{r}$

Step 3: Minimize the surface area

To minimize $A$, we take the derivative of $A$ with respect to $r$, set it to zero, and solve for $r$.

$\frac{dA}{dr} = 2\pi r - \frac{324}{r^2}$ Set $\frac{dA}{dr} = 0$: $2\pi r = \frac{324}{r^2}$ Multiply both sides by $r^2$: $2\pi r^3 = 324$ Solve for $r$: $r^3 = \frac{324}{2\pi} = \frac{162}{\pi}$ $r = \left( \frac{162}{\pi} \right)^{\frac{1}{3}}$

Let me calculate the value of $r$.The radius $r$ is approximately $3.72 \, \text{cm}$.

Step 4: Find the corresponding height $h$

Using the formula $h = \frac{162}{\pi r^2}$, we can calculate the height. Let me compute that next.The height $h$ is also approximately $3.72 \, \text{cm}$.

Step 5: Calculate the minimum surface area

Now, let's calculate the surface area using the formula: $A = \pi r^2 + \frac{324}{r}$ Let me compute the minimum surface area.The minimum surface area of the open-top hollow cylinder is approximately $131 \, \text{cm}^2$.

Would you like further details or have any questions?

Here are 5 related questions you might consider:

1. How does the surface area change with different cylinder heights?
2. What would the surface area be if the cylinder had a closed top?
3. How do you derive the optimal dimensions for a different volume?
4. Can this method be applied to minimize the surface area of other shapes?
5. How does the ratio between height and radius affect the efficiency of material usage?

Tip: When solving optimization problems involving geometry, always check if taking derivatives helps find minimum or maximum values efficiently.