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:
\[ \frac{dA}{dr} \space = \space 2 \pi r \space – \space \frac{40000}{r^2}\]
\[ 0 \space = \space 2 \pi r \space – \space \frac{40000}{r^2}\]
\[\frac{4000}{r^2} \space = \space 2 \pi r\]
\[r^3 \space = \space \frac{2000}{\pi} \]
\[r \space = \space 8.6\]
\[h \space = \space 8.6\]