Cylinder Volume Relation to Surface Area

The question asks for the factor by which the curved surface area (CSA) of a right circular cylinder must be multiplied to get its volume (V).

Cylinder Formulas Recap

Let's recall the standard formulas for a right circular cylinder:

• Base Radius: $r$
• Height: $h$
• Volume (V): The space occupied by the cylinder. The formula is:

  $V = \pi r^2 h$

• Curved Surface Area (CSA): The area of the side surface (excluding the top and bottom circular bases). The formula is:

  $CSA = 2 \pi r h$

Calculating the Multiplier Factor

We are looking for a factor, let's call it $F$, such that:

$V = CSA \times F$

To find this factor $F$, we can rearrange the equation:

$F = \frac{V}{CSA}$

Now, substitute the formulas for $V$ and $CSA$:

$F = \frac{\pi r^2 h}{2 \pi r h}$

We can simplify this expression by cancelling out common terms in the numerator and the denominator:

• $\pi$ cancels out.
• One factor of $r$ cancels out ($r^2$ becomes $r$).
• $h$ cancels out.

After cancellation, the expression for $F$ becomes:

$F = \frac{r}{2}$

Conclusion

Therefore, the volume of a right circular cylinder is obtained by multiplying its curved surface area by $\frac{r}{2}$.

Comparing this result with the given options, we find that option 4 matches our calculated factor.