Question

A tank is in the shape of a cylinder with two hemispheres attached to both ends as shown in the figure. Its common diameter is 2 metres and total length is 8 metres. Find the total cost of painting the outer surface of this tank at the rate of 60 rupees per square metre . (Use $\pi =3.14$ as an approximation)

Answer 1

Hint: Here, we will use the formula for the curved surface area of a cylinder and a hemisphere to find the total surface area of this tank. The curved surface area of a cylinder of radius r and height h is given as $2\pi rh$ and that of a hemisphere of radius r is given as $2\pi {{r}^{2}}$. Then, we will multiply the total surface area with the rate per metre square to find the total cost.

Complete step-by-step answer:

We know that the formula for curved surface area of a cylinder is given as:

 $CSA=2\pi rh...........\left( 1 \right)$

Also, the curved surface area of a hemisphere is given as:

$CSA=2\pi {{r}^{2}}........\left( 2 \right)$

Here, it is given that the common diameter of the cylinder and the hemisphere is 2 m.

So, their common radius is $=\dfrac{2m}{2}=1m$.

The length of the cylinder is given 8 m.

On putting the given values for cylinder in equation (1), we get:

$\begin{align}

  & CSA=2\times 3.14\times 1\times 8{{m}^{2}} \\

 & \Rightarrow CSA=50.24{{m}^{2}} \\

\end{align}$

Similarly on putting the values for hemisphere in equation (2), we get:

$\begin{align}

  & CSA=2\times 3.14\times {{\left( 1m \right)}^{2}} \\

 & \Rightarrow CSA=6.28{{m}^{2}} \\

\end{align}$

So, the total surface area of the tank is:

$\begin{align}

  & =\text{CSA of cylinder + 2}\times \text{CSA of hemisphere } \\

 & =50.24{{m}^{2}}+2\times 6.28{{m}^{2}} \\

 & =50.24{{m}^{2}}+12.56{{m}^{2}} \\

 & =62.8{{m}^{2}} \\

\end{align}$

Since, the price for painting is Rs. 60 per metre square.

So, total cost for painting the tank is:

\[\begin{align}

  & =\text{Rs}.60\times \text{ total surface area of tank} \\

 & \text{=Rs}\text{.60}\times \text{62}\text{.8}{{\text{m}}^{2}} \\

 & =\text{Rs}\text{.3,768} \\

\end{align}\]

Hence, the cost for painting is Rs. 3,768.

Note: Students should note here that while calculating the total surface area of the solid, we multiply the CSA of hemisphere by 2 because there are two hemispheres at the ends. The calculations must be done properly to avoid mistakes.