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What are the dimensions of the lightest open-top right circular cylinder can hold a volume of 1000 cm^3 ?

The main objective of this question is to find the dimension of the open cylinder which has a volume of 1000 cm^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:

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

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

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

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

\[r \space = \space 8.6\]

\[h \space = \space 8.6\]

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