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The TSA of cylinder is $231c{{m}^{2}}$ if the CSA is ${{\dfrac{2}{3}}^{rd}}$of the TSA. Find radius and height.

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Last updated date: 13th Jun 2024
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Answer
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Hint: We will be using the concepts of mensuration to solve the problem. We will be using the formula to find the curved surface area and total surface area of the cylinder and form equations according to the conditions given in the question to find radius and height.

Complete step by step answer:
Now, we have been given that the total surface area of the cylinder is $231c{{m}^{2}}$.
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Now, we let the radius r and height h of the cylinder. Now, we know that the total surface area of a cylinder is \[2\pi rh+2\pi {{r}^{2}}\].
So, according to the question we have,
\[2\pi rh+2\pi {{r}^{2}}=231........\left( 1 \right)\]
Now, we know that the curved surface area of a cylinder is $2\pi rh$. Now, according to the question it is equal to ${{\dfrac{2}{3}}^{rd}}$of the TSA. Therefore,
\[\begin{align}
  & \Rightarrow 2\pi rh=\dfrac{2}{3}\left( 231 \right) \\
 & \Rightarrow 2\pi rh=2\times 77 \\
 & \Rightarrow \pi rh=77 \\
 & \Rightarrow \dfrac{22}{7}\times rh=77 \\
 & \Rightarrow rh=\dfrac{77\times 7}{22} \\
 & \Rightarrow rh=\dfrac{7\times 7}{2}...........\left( 2 \right) \\
\end{align}\]
Now, we put the value of $2\pi rh$ in (1). So, we have,
\[\begin{align}
  & \Rightarrow 154+2\pi {{r}^{2}}=231 \\
 & \Rightarrow 2\pi {{r}^{2}}=231-154 \\
 & \Rightarrow 2\pi {{r}^{2}}=77 \\
 & \Rightarrow 2\times \dfrac{22}{7}{{r}^{2}}=77 \\
 & \Rightarrow {{r}^{2}}=\dfrac{77\times 7}{2\times 22} \\
 & \Rightarrow {{r}^{2}}=\dfrac{7\times 7}{2\times 2} \\
 & \Rightarrow r=\dfrac{7}{2}cm \\
\end{align}\]
Now, we put $r=\dfrac{7}{2}$ in (2). So,
$\begin{align}
  & \dfrac{7}{2}\times h=\dfrac{7}{2}\times 7 \\
 & \Rightarrow h=7cm \\
\end{align}$

Therefore, the radius is $\dfrac{7}{2}cm$ and height is 7cm.

Note: To solve these types of questions it is important to note that we have first converted the data given in the question to equation and then solve them to find the answer. Also it has to be noted that there is a difference between total surface area and curved surface area that total surface area includes the area of the base and top of the cylinder as well but in curved surface we include the area of only curved surface.