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For Question 3(b), where you're asked to systematically vary the values of \( S \) (surface area) to determine the effect on the volume function, you provided three values: \( S = 100 \), \( S = 110 \), and \( S = 120 \). Here's how to approach it:

1. **Surface Area Formula**: 
   The total surface area \( S \) of a cylindrical can is given by:
   \[
   S = 2\pi r^2 + 2\pi r h
   \]
   Solving for \( h \), we get:
   \[
   h = \frac{S - 2\pi r^2}{2\pi r}
   \]
   
2. **Volume Formula**: 
   The volume \( V \) of the cylinder is:
   \[
   V = \pi r^2 h
   \]
   Substituting \( h \) into this expression for volume:
   \[
   V = \pi r^2 \left( \frac{S - 2\pi r^2}{2\pi r} \right)
   \]
   Simplifying further:
   \[
   V = \frac{r(S - 2\pi r^2)}{2}
   \]
   This is the volume as a function of the radius \( r \).

3. **Substitute Values of \( S \)**: 
   Now, let's substitute \( S = 100 \), \( S = 110 \), and \( S = 120 \) into the volume equation:

   For \( S = 100 \):
   \[
   V = \frac{r(100 - 2\pi r^2)}{2}
   \]

   For \( S = 110 \):
   \[
   V = \frac{r(110 - 2\pi r^2)}{2}
   \]

   For \( S = 120 \):
   \[
   V = \frac{r(120 - 2\pi r^2)}{2}
   \]

4. **Effect on Volume Function**: 
   By analyzing and graphing these functions, you will notice how changes in the surface area \( S \) impact the volume. Typically, increasing \( S \) allows for a larger maximum volume as the can size (radius and height) increases.

Would you like me to help plot these graphs or further elaborate on specific steps? Here are some follow-up questions for a deeper understanding:

1. What is the maximum volume for each value of \( S \)?
2. How does changing \( S \) shift the position of the maximum volume?
3. Can you calculate the radius that yields the maximum volume for \( S = 100 \), \( S = 110 \), and \( S = 120 \)?
4. How does the height of the can change with the surface area?
5. What happens to the volume when the radius becomes too large or too small for a fixed \( S \)?

**Tip**: When optimizing volume with constraints like surface area, always double-check your critical points and boundary conditions.

Explore how varying the surface area of a cylindrical can affects its volume. Learn key formulas such as the surface area and volume of a cylinder, and discover how to find the maximum volume through systematic variation of surface areas (S = 100, S = 110, and S = 120).

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