The main objective of this question is to find the dimension of the **open cylinder** which has a **volume** of **1000 c****m^3**.

This question uses the concept of the **volume and surface area** for the **circular cylinder** which is **open-top or close-top**. **Mathematically,** the volume of a **circular cylinder** is represented as:

\[V\space = \space \pi r^2h\]

**Where** $r$ is the **radius** while $h$ is the **height**.

## Expert Answer

In this question, we are **required** to find the **dimension** of the **open cylinder** which has a **volume** of $1000 cm^3$. **Mathematically,** the **volume** of a **circular right cylinder** is represented as:

\[V\space = \space \pi r^2h\]

**Where** $r$ is the **radius** while $h$ is the **height**.

If the **cylinder is close-top,** then **mathematically** the **surface area** of the **close-top cylinder** is represented by:

\[V\space = \space 2\pi r^2 \space + \space 2\pi rh\]

And if the cylinder is **open-top,** then **mathematically** the **surface area** of the **open-top cylinder** is represented by:

\[V\space = \space \pi r^2 \space + \space 2\pi rh\]

**So**:

\[ \pi r^2h \space = \space 1000 \]

**Dividing** by $\pi r^2$ results in:

\[h \space = \space \frac{1000}{ \pi r^2h}\]

\[A \space = \space \pi r^2 \space + \space 2 \pi r (\frac{1000}{ \pi r^2})\]

\[= \space \pi r^2 \space + \space \frac{2000}{r}\]

**Taking** the **derivative** of $A$ with **respect** to $r$ **results** in:

\[ \frac{dA}{dr} \space = \space 2 \pi r \space – \space \frac{20000}{r^2}\]

\[ 0 \space = \space 2 \pi r \space – \space \frac{20000}{r^2}\]

\[\frac{2000}{r^2} \space = \space 2 \pi r\]

**Dividing** by $r$ results in:

\[r^3 \space = \space \frac{1000}{\pi} \]

**Simplifying** for $r$ will result in:

\[r \space = \space 6.83\]

**Hence** $r$ = $h$ = $ 6.83$.

## Numerical Results

The **dimensions** of **open-top cylinder** which can hold a **volume** of $1000 cm^3$ is $r = h= 6.83$.

## Example

Find the dimension of the open cylinder which has a volume of 2000 c m^3.

In this question, we are required to find the **dimension** of the **open cylinder** which has a **volume** of $2000 cm^3$. **Mathematically,** the **volume** of a **circular right cylinder** is represented as:

\[V\space = \space \pi r^2h\]

Where $r$ is the **radius** while $h$ is the **height**.

If the cylinder is **close-top,** then **mathematically** the surface area of the **close-top cylinder** is represented by:

\[V\space = \space 2\pi r^2 \space + \space 2\pi rh\]

And if the **cylinder** is **open-top,** then **mathematically** the **surface area** of the **open-top cylinder** is represented by:

\[V\space = \space \pi r^2 \space + \space 2\pi rh\]

\[ \pi r^2h \space = \space 2000 \]

\[h \space = \space \frac{2000}{ \pi r^2h}\]

\[A \space = \space \pi r^2 \space + \space 2 \pi r (\frac{2000}{ \pi r^2})\]

\[= \space \pi r^2 \space + \space \frac{4000}{r}\]

**Taking** the **derivative** of $A$ with respect to $r$ results in:

\[ \frac{dA}{dr} \space = \space 2 \pi r \space – \space \frac{40000}{r^2}\]

\[ 0 \space = \space 2 \pi r \space – \space \frac{40000}{r^2}\]

\[\frac{4000}{r^2} \space = \space 2 \pi r\]

\[r^3 \space = \space \frac{2000}{\pi} \]

\[r \space = \space 8.6\]

\[h \space = \space 8.6\]