\( \frac{\text { Volume of cylinder }}{\text { surface area of cylinder }} \)

To find the ratio of the **volume** of a cylinder to its **surface area**, let's start by defining the cylinder's dimensions: - **Radius (r)** - **Height (h)** ### Volume of a Cylinder The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] ### Surface Area of a Cylinder The surface area \( S \) of a cylinder includes the areas of the two circular bases and the rectangular side (which wraps around to form the lateral surface). It is calculated as: \[ S = 2\pi r^2 + 2\pi r h = 2\pi r (r + h) \] - \( 2\pi r^2 \) accounts for the top and bottom circular areas. - \( 2\pi r h \) accounts for the lateral surface area. ### Ratio of Volume to Surface Area To find the ratio \( \frac{V}{S} \), divide the volume by the surface area: \[ \frac{V}{S} = \frac{\pi r^2 h}{2\pi r (r + h)} \] Simplify the expression by canceling out common terms: \[ \frac{V}{S} = \frac{r h}{2(r + h)} \] ### Final Expression So, the ratio of the volume of a cylinder to its surface area is: \[ \frac{\text{Volume}}{\text{Surface Area}} = \frac{r h}{2(r + h)} \] ### Interpretation This ratio provides a relationship between the cylinder's dimensions. For example: - **If the height \( h \) increases while the radius \( r \) remains constant**, the ratio increases, indicating that the volume grows faster relative to the surface area. - **If the radius \( r \) increases while the height \( h \) remains constant**, the ratio also increases, but the impact depends on the relative sizes of \( r \) and \( h \). Understanding this ratio is useful in various applications, such as optimizing materials usage for manufacturing cylinders where maximizing volume while minimizing surface area (or vice versa) is desired.