To solve this problem, let's break down the requirements and calculations.

Problem Setup

1. Given: Total surface area of a cylindrical can is $170 \, \text{cm}^2$.
2. Objective: Maximize the volume $V$ of the cylinder.
3. Formulas:
   • Surface Area: $S = 2\pi r^2 + 2\pi r h = 170$
   • Volume: $V = \pi r^2 h$

Steps

1. Express $h$ in terms of $r$ using the surface area equation: $2\pi r^2 + 2\pi r h = 170$ Solve for $h$: $h = \frac{170 - 2\pi r^2}{2\pi r}$

2. Substitute $h$ into the volume formula: $V = \pi r^2 \left( \frac{170 - 2\pi r^2}{2\pi r} \right)$ Simplify this expression to get $V$ in terms of $r$ only.

3. Differentiate $V$ with respect to $r$ and set to zero to find the value of $r$ that maximizes the volume.

4. Substitute the value of $r$ back into the equation for $h$ to find the corresponding height.

Would you like me to delve into the detailed calculations and solve for the exact values of $r$ and $h$?

1. How do you find the surface area of a cylinder?
2. What is the relationship between surface area and volume?
3. Why do we differentiate to find maximum volume?
4. How do we interpret the result in terms of physical dimensions?
5. What units should be used for volume?

Tip: When optimizing with constraints, always express one variable in terms of another to simplify calculations.