The total surface area of a solid right circular cylinder is 1540 $c{{m}^{2}}$. If the height is four times the radius of the base, then find the height of the cylinder.

- Hint:In order to solve this question, we should know that the total surface area of a cylinder is calculated by using the formula, $2\pi r\left( r+h \right)$. So, we will assume h as the height and r as the radius and from the given conditions, we will form 2 equations of 2 variables and then by substitution method, we will find the height of the cylinder.

Complete step-by-step solution -

In this question, we have been asked to find the height of a cylinder which has total surface area of 1540 $c{{m}^{2}}$ and the height of the cylinder is four times the radius of the base. So, to solve this question, we will first draw the cylinder. So, we get,

Now, let us consider the height of the cylinder as ‘h’ and the radius of the base of the cylinder as ‘r’. So, according to the given condition, that is the height of the cylinder is four times the radius of the base, we can write,
h = 4r ……… (i)
Now, we know that the total surface area of a cylinder is given by the formula, $2\pi r\left( r+h \right)$ and we have been given that the total surface area of the cylinder is 1540 $c{{m}^{2}}$. So, we can write
$2\pi r\left( r+h \right)=1540.........\left( ii \right)$
Now, from equation (i), we also get $r=\dfrac{h}{4}$. So, we will put the value of r in equation (ii), so we will get,
$2\pi \dfrac{h}{4}\left( \dfrac{h}{4}+h \right)=1540$
Now, we will simplify it further to get the answer. So, we get,
\[\begin{align}
  & 2\pi \dfrac{{{h}^{2}}}{16}+2\pi \dfrac{{{h}^{2}}}{4}=1540 \\
 & 2\pi \dfrac{{{h}^{2}}}{4}\left( \dfrac{1}{4}+1 \right)=1540 \\
 & 2\pi \dfrac{{{h}^{2}}}{4}\times \dfrac{5}{4}=1540 \\
 & {{h}^{2}}=\dfrac{1540\times 4\times 4}{2\pi \times 5} \\
 & {{h}^{2}}=\dfrac{1540\times 16}{10\pi } \\
\end{align}\]
Now, we know that $\pi =\dfrac{22}{7}$. So, we can write,
$\begin{align}
  & {{h}^{2}}=\dfrac{1540\times 16}{10\times \dfrac{22}{7}} \\
 & {{h}^{2}}=\dfrac{1540\times 16\times 7}{10\times 22}=784 \\
 & h=\sqrt{784} \\
 & h=28 \\
\end{align}$
Hence, we get the height of the cylinder as 28 cm.

Note: While solving this question, we need to remember that $\pi =\dfrac{22}{7}$ and we must remember the formula of total surface area of the cylinder, that is, $2\pi r\left( r+h \right)$. And there are possibilities that we might make a calculation mistake because this question contains a lot of calculations. Also, we can find the value of h by finding the value of r first and then finding the value of h, but that will be time consuming and will make the solution more complicated.