$$ \frac{ ext { Volume of cylinder }}{ ext { surface area of cylinder }} $$

Question

$$ \frac{	ext { Volume of cylinder }}{	ext { surface area of cylinder }} $$

UpStudy AI · Accepted Answer

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.
The ratio of the volume of a cylinder to its surface area is $$ \frac{r \times h}{2(r + h)} $$.

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Solución

Volume of a Cylinder

Surface Area of a Cylinder

Ratio of Volume to Surface Area

Final Expression

Interpretation

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