Question

Find the curved surface area of cylinder where the ratio between the radius of the base and the height of cylindrical is 2:3 and its volume is$1617c{{m}^{3}}$. A cylindrical pillar has volume $\text{924}{{\text{m}}^{3}}$ ad curved surface area $264{{\text{m}}^{2}}$. Find the height and diameter. Find the ratio of their radii of the cylinder to the pillar.\[\]

Answer 1

Hint: We assume the height of the cylinder as ${{h}_{1}}$, the radius of the base of cylinder as ${{r}_{1}}$, the height of the cylindrical pillar as ${{h}_{2}}$ and the radius of the base of the cylindrical pillar as ${{r}_{2}}$. We use the formula for curved surface area $A=2\pi rh$ and volume $V=\pi {{r}^{2}}h$ for any cylinder with height $h$ and radius $r$ to obtain the asked quantities and ${{r}_{1}},{{r}_{2}}$. We find ${{r}_{1}}:{{r}_{2}}$.\[\]

Complete step by step answer:
We know that a cylinder is a solid three dimensional objects with one curved surface and two circular plane surfaces called bases. The height of the cylinder is denoted as $h$ and the radius at the base as $r$ . The curved surface area $A$ of the cylinder is given by
\[A=\pi rh\]
The volume of the cylinder is given by
\[V=\pi {{r}^{2}}h\]
Let assume the height of the cylinder as ${{h}_{1}}$ and the radius of the cylinder as ${{r}_{1}}$. We are given in the question that the ratio between the radius of the base and the height of cylindrical is 2:3 which means
\[\begin{align}
  & {{r}_{1}}:{{h}_{1}}=2:3 \\
 & \Rightarrow \dfrac{{{r}_{1}}}{{{h}_{1}}}=\dfrac{2}{3} \\
 & \Rightarrow {{h}_{1}}=\dfrac{3}{2}{{r}_{1}} \\
\end{align}\]
We are also given in the question that the volume $V$ of the cylinder is $1617c{{m}^{3}}$. We us the formula for volume of cylinder with $r={{r}_{1}},h={{h}_{1}}$ and have,
\[\begin{align}
  & V=1617 \\
 & \Rightarrow \pi {{r}_{1}}^{2}{{h}_{1}}=1617 \\
\end{align}\]
We put previously obtained \[{{h}_{1}}=\dfrac{3}{2}{{r}_{1}}\] and proceed to get,
\[\begin{align}
  & \Rightarrow \dfrac{22}{7}\times {{r}_{1}}\times {{r}_{1}}\times \dfrac{3}{2}=1617 \\
 & \Rightarrow {{r}_{1}}^{3}=343 \\
 & \Rightarrow {{r}_{1}}=7 \\
\end{align}\]
So the radius of the cylinder is ${{r}_{1}}=7$cm. We find the curved surface area $A$ of the cylinder in $\text{c}{{\text{m}}^{2}}$as,
\[A=2\pi {{r}_{1}}{{h}_{1}}=2\times \pi \times {{r}_{1}}\times {{r}_{1}}\times \dfrac{3}{2}=2\times \dfrac{22}{7}\times 7\times 7\times \dfrac{3}{2}=462\]
We are also given that the cylindrical pillar has volume $\text{924}{{\text{m}}^{3}}$ ad curved surface area $264{{\text{m}}^{2}}$. Let assume the height of the cylindrical pillar as ${{h}_{2}}$ and the radius of the cylindrical pillar as ${{r}_{2}}$.So we have from the formula of volume of cylinder,
\[\begin{align}
  & V=\pi {{r}_{2}}^{2}{{h}_{2}}=924 \\
 & \Rightarrow \left( \pi {{r}_{2}}{{h}_{2}} \right){{r}_{2}}=924...(1) \\
\end{align}\]
We have from the formula for curved surface area,
\[\begin{align}
  & A=2\pi {{r}_{2}}{{h}_{2}}=264 \\
 & \Rightarrow \pi {{r}_{2}}{{h}_{2}}=\dfrac{264}{2}=132..(2) \\
\end{align}\]
We put the value in equation(1) and get,
\[\begin{align}
  & \left( \pi {{r}_{2}}{{h}_{2}} \right){{r}_{2}}=924 \\
 & \Rightarrow 132{{r}_{2}}=924 \\
 & \Rightarrow {{r}_{2}}=\dfrac{924}{132}=7m \\
\end{align}\]
We know that the diameter is twice the length of radius which $2{{r}_{2}}=2\times 7=14$m. We put ${{r}_{2}}$in equation (2) to get,
\[\begin{align}
  & \dfrac{22}{7}\times 7\times {{h}_{2}}=132 \\
 & \Rightarrow {{h}_{2}}=6\text{m} \\
\end{align}\]
 The ratio of the radii of the cylinder and the cylindrical pillar is
\[{{r}_{1}}:{{r}_{2}}=\dfrac{7cm}{7m}=\dfrac{7cm}{700cm}=\dfrac{1}{100}=1:100\]

Note:
We have used the value $\pi =\dfrac{22}{7}$ because the volumes and areas given in the question were multiples of 7. We also have to be careful that ${{r}_{1}},{{r}_{2}}$ are not the same units of length and we needed to convert them into the same units. The total surface area of cylinder is $2\pi \left( rh+{{r}^{2}} \right)$.