Question

A storage tank consists of a circular cylinder with a hemisphere stuck in either end. If the external diameter of the cylinder be $1.4\;m$ and its length be $8\;m$. Find the cost of painting it on the outside at a rate of $20$ per ${m^2}$.

Answer 1

In this problem it is given a storage tank consisting of a circular cylinder with a hemisphere stuck in either end. If the external diameter of the cylinder be $1.4\;m$ and its length be $8\;m$. We have to find the cost of painting it on the outside at $20$ per ${m^2}$. First we have to find the radius of the cylinder with the help of given diameter and then find the total surface area of the cylinder by using the formula of surface area of the cylinder.

Complete step by step solution:
It is given that a storage tank consists of a circular cylinder with a hemisphere stuck in either end. If the external diameter of the cylinder be $1.4\;m$ and its length be $8\;m$.
We have to find the cost of painting it on the outside at $20$ per ${m^2}$.
Since, it is given the external diameter of the cylinder is $1.4\;m$.
To find the radius of the cylinder we have to divide the diameter of the cylinder by two. because it is known that the half of the diameter is known as radius.

Therefore, we have
The radius of the cylinder$= \dfrac{{{\text{Diameter of the cylinder}}}}{2}$
Substituting the given value of the diameter of the cylinder which is $1.4\;m$ in the above expression.
The radius of the cylinder$\begin{array}{l} = \dfrac{{1.4\;m}}{2}\\\ = 0.7\;m\end{array}$
Let, radius of the cylinder is denoted by $r$
Thus, we have $r = 0.7\;m$
Also, given the length of the cylinder is $8\;m$ which is nothing but the height of the cylinder.
Let, height of the cylinder is denoted by $h$
Thus, we have $h = 8\;m$.
Using the formula of the total surface area of the cylinder.
The total surface area of a cylinder is $\begin{array}{l} = 2\pi {r^2} + 2\pi rh\\\ = 2\pi r\left( {r + h} \right)\end{array}$.
Where, $r$ is the radius of the cylinder and $h$ is the height of the cylinder respectively.
Also, it is known that the standard value of the $\pi$ is $\dfrac{{22}}{7}$.

Now,
Total surface area of cylinder$= 2\pi r\left( {r + h} \right)$
Substituting the value of the $\pi = \dfrac{{22}}{7}$ , radius of the cylinder $r = 0.7\;m$ and height of the cylinder $h = 8\;m$ in the above equation.
Total surface area of cylinder$\begin{array}{l} = 2 \times \dfrac{{22}}{7} \times \left( {0.7\;m} \right) \times \left( {0.7\;m + 8\;m} \right)\\\ = 2 \times \dfrac{{22}}{7} \times 0.7 \times 8.7\;{m^2}\end{array}$
Since, it is given the cost of the painting on the outside at a rate of $20$ per ${m^2}$.
So, by substituting the rate of painting per ${m^2}$ which is $20$ in the total surface area of the cylinder we found the required cost of painting.
Therefore, the required cost of painting is \begin{array}{l} = Rs.\left\\{ {2 \times \dfrac{{22}}{7} \times 0.7 \times 8.7 \times 20} \right\\}\\\ = Rs.765.60\end{array}.

Hence, the cost of painting is $Rs.765.60$

Note: Here, given the diameter of the cylinder and height of the cylinder and we have to find the cost of painting at a given rate per ${m^2}$. First, we find the radius of the cylinder with the help of a given diameter and then using the formula of the total surface area of the cylinder. Finally, we substitute the given rate at the total surface area of the cylinder and we get our results.